(*^
::[	Information =

	"This is a Mathematica Notebook file.  It contains ASCII text, and can be
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	received this as email, use your mail application or copy/paste to save 
	everything from the line containing (*^ down to the line containing ^*)
	into a plain text file.  On some systems you may have to give the file a 
	name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
	The line below identifies what version of Mathematica created this file,
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:[font = input; initialization; preserveAspect]
*)
Exercises on Systems of Differential Equations
(*
:[font = input; initialization; preserveAspect]
*)
$Post:=If[MatrixQ[#],MatrixForm[#],#]&
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Table[i+j,{i,4},{j,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 3, 4, 5}, {3, 4, 5, 6}, {4, 5, 6, 7}, 
 
   {5, 6, 7, 8}}]
;[o]
2   3   4   5

3   4   5   6

4   5   6   7

5   6   7   8
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Eigensystem[M][[2]]//Together]; P//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{2, 1, (-145 + 27*30^(1/2))/(-265 + 48*30^(1/2)), 
 
    (145 + 27*30^(1/2))/(265 + 48*30^(1/2))}, 
 
   {-3, -2, (-185 + 34*30^(1/2))/(-265 + 48*30^(1/2)), 
 
    (185 + 34*30^(1/2))/(265 + 48*30^(1/2))}, 
 
   {0, 1, (-225 + 41*30^(1/2))/(-265 + 48*30^(1/2)), 
 
    (225 + 41*30^(1/2))/(265 + 48*30^(1/2))}, {1, 0, 1, 1}}]
;[o]
          -145 + 27 Sqrt[30]   145 + 27 Sqrt[30]
          ------------------   -----------------
2    1    -265 + 48 Sqrt[30]   265 + 48 Sqrt[30]

          -185 + 34 Sqrt[30]   185 + 34 Sqrt[30]
          ------------------   -----------------
-3   -2   -265 + 48 Sqrt[30]   265 + 48 Sqrt[30]

          -225 + 41 Sqrt[30]   225 + 41 Sqrt[30]
          ------------------   -----------------
0    1    -265 + 48 Sqrt[30]   265 + 48 Sqrt[30]

1    0    1                    1
:[font = input; initialization; preserveAspect; startGroup]
*)
T=(Inverse[P]//Together).M.P//Together;
T//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{0, 0, 0, 0}, {0, 0, 0, 0}, 
 
   {0, 0, 2*(5 - 30^(1/2)), 0}, {0, 0, 0, 2*(5 + 30^(1/2))}}]
;[o]
0   0   0                  0

0   0   0                  0

0   0   2 (5 - Sqrt[30])   0

0   0   0                  2 (5 + Sqrt[30])
:[font = input; initialization; preserveAspect; startGroup]
*)
z=Table[Z[i][x]/.DSolve[Z[i]'[x]==T[[i,i]]Z[i][x],Z[i][x],x] \
                                        /.{C[1]->K[i]},{i,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{K[1]}, {K[2]}, {E^((10 - 2*30^(1/2))*x)*K[3]}, 
 
   {E^((10 + 2*30^(1/2))*x)*K[4]}}]
;[o]

K[1]


K[2]

 (10 - 2 Sqrt[30]) x
E                    K[3]

 (10 + 2 Sqrt[30]) x
E                    K[4]
:[font = input; initialization; preserveAspect; startGroup]
*)
y=P.z//Together//Simplify
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(34*K[1] + 17*K[2] - 
 
       7*E^((10 - 2*30^(1/2))*x)*K[3] - 
 
       3*30^(1/2)*E^((10 - 2*30^(1/2))*x)*K[3] - 
 
       7*E^(2*(5 + 30^(1/2))*x)*K[4] + 
 
       3*30^(1/2)*E^(2*(5 + 30^(1/2))*x)*K[4])/17}, 
 
   {(-51*K[1] - 34*K[2] + E^((10 - 2*30^(1/2))*x)*K[3] - 
 
       2*30^(1/2)*E^((10 - 2*30^(1/2))*x)*K[3] + 
 
       E^(2*(5 + 30^(1/2))*x)*K[4] + 
 
       2*30^(1/2)*E^(2*(5 + 30^(1/2))*x)*K[4])/17}, 
 
   {(17*K[2] + 9*E^((10 - 2*30^(1/2))*x)*K[3] - 
 
       30^(1/2)*E^((10 - 2*30^(1/2))*x)*K[3] + 
 
       9*E^(2*(5 + 30^(1/2))*x)*K[4] + 
 
       30^(1/2)*E^(2*(5 + 30^(1/2))*x)*K[4])/17}, 
 
   {K[1] + E^((10 - 2*30^(1/2))*x)*K[3] + 
 
     E^(2*(5 + 30^(1/2))*x)*K[4]}}]
;[o]
                        (10 - 2 Sqrt[30]) x
(34 K[1] + 17 K[2] - 7 E                    K[3] - 
 
                 (10 - 2 Sqrt[30]) x
     3 Sqrt[30] E                    K[3] - 
 
        2 (5 + Sqrt[30]) x
     7 E                   K[4] + 
 
                 2 (5 + Sqrt[30]) x
     3 Sqrt[30] E                   K[4]) / 17

                       (10 - 2 Sqrt[30]) x
(-51 K[1] - 34 K[2] + E                    K[3] - 
 
                (10 - 2 Sqrt[30]) x
    2 Sqrt[30] E                    K[3] + 
 
     2 (5 + Sqrt[30]) x
    E                   K[4] + 
 
                2 (5 + Sqrt[30]) x
    2 Sqrt[30] E                   K[4]) / 17

              (10 - 2 Sqrt[30]) x
(17 K[2] + 9 E                    K[3] - 
 
              (10 - 2 Sqrt[30]) x
    Sqrt[30] E                    K[3] + 
 
       2 (5 + Sqrt[30]) x
    9 E                   K[4] + 
 
              2 (5 + Sqrt[30]) x
    Sqrt[30] E                   K[4]) / 17

        (10 - 2 Sqrt[30]) x         2 (5 + Sqrt[30]) x
K[1] + E                    K[3] + E                   K[4]
:[font = input; initialization; preserveAspect; startGroup]
*)
y//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0.05882352941176470589*
 
     (34.*K[1.] + 17.*K[2.] - 
 
       (23.4316767251549834*K[3.])/
 
        2.718281828459045235^(0.9544511501033222691*x) + 
 
       9.431676725154983404*
 
        2.718281828459045235^(20.95445115010332227*x)*K[4.])}
 
    , {0.05882352941176470589*
 
     (-51.*K[1.] - 34.*K[2.] - 
 
       (9.954451150103322269*K[3.])/
 
        2.718281828459045235^(0.9544511501033222691*x) + 
 
       11.95445115010332227*
 
        2.718281828459045235^(20.95445115010332227*x)*K[4.])}
 
    , {0.05882352941176470589*
 
     (17.*K[2.] + (3.522774424948338865*K[3.])/
 
        2.718281828459045235^(0.9544511501033222691*x) + 
 
       14.47722557505166113*
 
        2.718281828459045235^(20.95445115010332227*x)*K[4.])}
 
    , {K[1.] + K[3.]/
 
      2.718281828459045235^(0.9544511501033222691*x) + 
 
     2.718281828459045235^(20.95445115010332227*x)*K[4.]}}]
;[o]
                                     23.4317 K[3.]
0.0588235 (34. K[1.] + 17. K[2.] - ----------------- + 
                                          0.954451 x
                                   2.71828
 
                   20.9545 x
    9.43168 2.71828          K[4.])

                                      9.95445 K[3.]
0.0588235 (-51. K[1.] - 34. K[2.] - ----------------- + 
                                           0.954451 x
                                    2.71828
 
                   20.9545 x
    11.9545 2.71828          K[4.])

                         3.52277 K[3.]
0.0588235 (17. K[2.] + ----------------- + 
                              0.954451 x
                       2.71828
 
                   20.9545 x
    14.4772 2.71828          K[4.])

              K[3.]                20.9545 x
K[1.] + ----------------- + 2.71828          K[4.]
               0.954451 x
        2.71828
:[font = input; initialization; preserveAspect]
*)
Solution check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
D[y,x]-M.y//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0}, {0}, {0}, {0}}]
;[o]
0

0

0

0
:[font = input; initialization; preserveAspect]
*)
Exercise 2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Table[Mod[i+j,2],{i,4},{j,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1, 0, 1}, {1, 0, 1, 0}, {0, 1, 0, 1}, 
 
   {1, 0, 1, 0}}]
;[o]
0   1   0   1

1   0   1   0

0   1   0   1

1   0   1   0
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Eigensystem[M][[2]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1, 0, -1, 1}, {1, -1, 0, 1}, {-1, 0, 1, 1}, 
 
   {1, 1, 0, 1}}]
;[o]
-1   0    -1   1

1    -1   0    1

-1   0    1    1

1    1    0    1
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].M.P
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-2, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, 
 
   {0, 0, 0, 2}}]
;[o]
-2   0    0    0

0    0    0    0

0    0    0    0

0    0    0    2
:[font = input; initialization; preserveAspect; startGroup]
*)
z=Table[Z[i][x]/.DSolve[Z[i]'[x]==T[[i,i]]Z[i][x],Z[i][x],x] \
                                        /.{C[1]->K[i]},{i,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{K[1]/E^(2*x)}, {K[2]}, {K[3]}, {E^(2*x)*K[4]}}]
;[o]
K[1]
----
 2 x
E




K[2]




K[3]



 2 x
E    K[4]
:[font = input; initialization; preserveAspect; startGroup]
*)
y=P.z
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-(K[1]/E^(2*x)) - K[3] + E^(2*x)*K[4]}, 
 
   {K[1]/E^(2*x) - K[2] + E^(2*x)*K[4]}, 
 
   {-(K[1]/E^(2*x)) + K[3] + E^(2*x)*K[4]}, 
 
   {K[1]/E^(2*x) + K[2] + E^(2*x)*K[4]}}]
;[o]
  K[1]            2 x
-(----) - K[3] + E    K[4]
   2 x
  E

K[1]           2 x
---- - K[2] + E    K[4]
 2 x
E

  K[1]            2 x
-(----) + K[3] + E    K[4]
   2 x
  E

K[1]           2 x
---- + K[2] + E    K[4]
 2 x
E
:[font = input; initialization; preserveAspect; startGroup]
*)
D[y,x]-M.y//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0}, {0}, {0}, {0}}]
;[o]
0

0

0

0
:[font = input; initialization; preserveAspect]
*)
Exercise 3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M={{1,-2,-2,-2,-2},   \
   {2,6,4,4,4},       \
   {-3,-6,-6,-10,-10},\
   {1,2,3,6,1},       \
   {1,2,3,4,9}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, -2, -2, -2, -2}, {2, 6, 4, 4, 4}, 
 
   {-3, -6, -6, -10, -10}, {1, 2, 3, 6, 1}, {1, 2, 3, 4, 9}}]
;[o]
1     -2    -2    -2    -2

2     6     4     4     4

-3    -6    -6    -10   -10

1     2     3     6     1

1     2     3     4     9
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Eigensystem[M][[2]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, -2, 1, 0, 0}, {0, 1, -2, 0, 0}, 
 
   {-2, 0, 1, 1, 0}, {1, 0, 0, -2, -1}, {0, 0, 0, 1, 1}}]
;[o]
2    -2   1    0    0

0    1    -2   0    0

-2   0    1    1    0

1    0    0    -2   -1

0    0    0    1    1
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].M.P
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 0, 0, 0, 0}, {0, 2, 0, 0, 0}, 
 
   {0, 0, 3, 0, 0}, {0, 0, 0, 4, 0}, {0, 0, 0, 0, 5}}]
;[o]
2   0   0   0   0

0   2   0   0   0

0   0   3   0   0

0   0   0   4   0

0   0   0   0   5
:[font = input; initialization; preserveAspect; startGroup]
*)
z=Table[Z[i][x]/.DSolve[Z[i]'[x]==T[[i,i]]Z[i][x],Z[i][x],x] \
                                        /.{C[1]->K[i]},{i,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{E^(2*x)*K[1]}, {E^(2*x)*K[2]}, {E^(3*x)*K[3]}, 
 
   {E^(4*x)*K[4]}, {E^(5*x)*K[5]}}]
;[o]
 2 x
E    K[1]

 2 x
E    K[2]

 3 x
E    K[3]

 4 x
E    K[4]

 5 x
E    K[5]
:[font = input; initialization; preserveAspect; startGroup]
*)
y=P.z
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2*E^(2*x)*K[1] - 2*E^(2*x)*K[2] + 
 
     E^(3*x)*K[3]}, {E^(2*x)*K[2] - 2*E^(3*x)*K[3]}, 
 
   {-2*E^(2*x)*K[1] + E^(3*x)*K[3] + E^(4*x)*K[4]}, 
 
   {E^(2*x)*K[1] - 2*E^(4*x)*K[4] - E^(5*x)*K[5]}, 
 
   {E^(4*x)*K[4] + E^(5*x)*K[5]}}]
;[o]
   2 x           2 x         3 x
2 E    K[1] - 2 E    K[2] + E    K[3]

 2 x           3 x
E    K[2] - 2 E    K[3]

    2 x         3 x         4 x
-2 E    K[1] + E    K[3] + E    K[4]

 2 x           4 x         5 x
E    K[1] - 2 E    K[4] - E    K[5]

 4 x         5 x
E    K[4] + E    K[5]
:[font = input; initialization; preserveAspect; startGroup]
*)
D[y,x]-M.y//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0}, {0}, {0}, {0}, {0}}]
;[o]
0

0

0

0

0
:[font = input; initialization; preserveAspect]
*)
Exercise 4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M={{2,0,-1,-1,-1},   \
   {3,8,8,7,7},      \
   {-3,-6,-6,-7,-8}, \
   {1,2,3,5,4},      \
   {0,0,0,0,2}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 0, -1, -1, -1}, {3, 8, 8, 7, 7}, 
 
   {-3, -6, -6, -7, -8}, {1, 2, 3, 5, 4}, {0, 0, 0, 0, 2}}]
;[o]
2    0    -1   -1   -1

3    8    8    7    7

-3   -6   -6   -7   -8

1    2    3    5    4

0    0    0    0    2
:[font = input; initialization; preserveAspect; startGroup]
*)
EV=Eigensystem[M][[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-3, 5, -5, 2, 0}, {-2, 1, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0}, {-1, -1, 1, 0, 0}, {0, 0, 0, 0, 0}}]
;[o]
-3   5    -5   2    0

-2   1    0    0    0

0    0    0    0    0

-1   -1   1    0    0

0    0    0    0    0
:[font = input; initialization; preserveAspect]
*)
Since we have only 3 linearly independent, nonzero eigenvectors,
M is not diagonalizable.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[EV]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0, 1}, {1, 2, 3, 4, 0}}]
;[o]
0   0   0   0   1

1   2   3   4   0
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Join[{EV[[1]]},{EV[[2]]},{EV[[4]]},%]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-3, -2, -1, 0, 1}, {5, 1, -1, 0, 2}, 
 
   {-5, 0, 1, 0, 3}, {2, 0, 0, 0, 4}, {0, 0, 0, 1, 0}}]
;[o]
-3   -2   -1   0    1

5    1    -1   0    2

-5   0    1    0    3

2    0    0    0    4

0    0    0    1    0
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].M.P
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, 0, 5/3, 11}, {0, 2, 0, -11/6, -5}, 
 
   {0, 0, 3, -1/6, -15}, {0, 0, 0, 2, 0}, {0, 0, 0, 1/6, 3}}]
;[o]
                        5
                        -
1       0       0       3       11

                          11
                        -(--)
0       2       0         6     -5

                          1
                        -(-)
0       0       3         6     -15



0       0       0       2       0

                        1
                        -
0       0       0       6       3
:[font = input; initialization; preserveAspect; startGroup]
*)
<<LinearAlgebra`Master`
B=SubMatrix[T,{4,4},{2,2}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 0}, {1/6, 3}}]
;[o]


2   0

1
-
6   3
:[font = input; initialization; preserveAspect; startGroup]
*)
EVB=Eigensystem[B][[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-6, 1}, {0, 1}}]
;[o]
-6   1

0    1
:[font = input; initialization; preserveAspect; startGroup]
*)
PB=Transpose[EVB]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-6, 0}, {1, 1}}]
;[o]
-6   0

1    1
:[font = input; initialization; preserveAspect; startGroup]
*)
Table[If[i<4 || j<4,IdentityMatrix[5][[i,j]],PB[[i-3,j-3]]], \
                                                {i,5},{j,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, 
 
   {0, 0, 1, 0, 0}, {0, 0, 0, -6, 0}, {0, 0, 0, 1, 1}}]
;[o]
1    0    0    0    0

0    1    0    0    0

0    0    1    0    0

0    0    0    -6   0

0    0    0    1    1
:[font = input; initialization; preserveAspect; startGroup]
*)
P=P.%
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-3, -2, -1, 1, 1}, {5, 1, -1, 2, 2}, 
 
   {-5, 0, 1, 3, 3}, {2, 0, 0, 4, 4}, {0, 0, 0, -6, 0}}]
;[o]
-3   -2   -1   1    1

5    1    -1   2    2

-5   0    1    3    3

2    0    0    4    4

0    0    0    -6   0
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].M.P
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, 0, 1, 11}, {0, 2, 0, 6, -5}, 
 
   {0, 0, 3, -14, -15}, {0, 0, 0, 2, 0}, {0, 0, 0, 0, 3}}]
;[o]
1     0     0     1     11

0     2     0     6     -5

0     0     3     -14   -15

0     0     0     2     0

0     0     0     0     3
:[font = input; initialization; preserveAspect; startGroup]
*)
z=Array[0&,5];
Do[z[[6-i]]=Z[6-i][x]/.DSolve[Z[6-i]'[x]==T[[6-i,6-i]]Z[6-i][x] \
                            +Sum[T[[6-i,j]]z[[j,1]],{j,7-i,5}], \
                            Z[6-i][x],x]/.{C[1]->K[6-i]},{i,5}]
z//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{E^x*K[1] + E^(2*x)*K[4] + (11*E^(3*x)*K[5])/2}, 
 
   {E^(2*x)*K[2] + 6*E^(2*x)*x*K[4] - 5*E^(3*x)*K[5]}, 
 
   {E^(3*x)*K[3] + 14*E^(2*x)*K[4] - 15*E^(3*x)*x*K[5]}, 
 
   {E^(2*x)*K[4]}, {E^(3*x)*K[5]}}]
;[o]
                          3 x
 x         2 x        11 E    K[5]
E  K[1] + E    K[4] + ------------
                           2



 2 x           2 x             3 x
E    K[2] + 6 E    x K[4] - 5 E    K[5]



 3 x            2 x            3 x
E    K[3] + 14 E    K[4] - 15 E    x K[5]



 2 x
E    K[4]



 3 x
E    K[5]
:[font = input; initialization; preserveAspect; startGroup]
*)
y=P.z//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-3*E^x*K[1] - 2*E^(2*x)*K[2] - E^(3*x)*K[3] - 
 
     16*E^(2*x)*K[4] - 12*E^(2*x)*x*K[4] - 
 
     (11*E^(3*x)*K[5])/2 + 15*E^(3*x)*x*K[5]}, 
 
   {5*E^x*K[1] + E^(2*x)*K[2] - E^(3*x)*K[3] - 
 
     7*E^(2*x)*K[4] + 6*E^(2*x)*x*K[4] + 
 
     (49*E^(3*x)*K[5])/2 + 15*E^(3*x)*x*K[5]}, 
 
   {-5*E^x*K[1] + E^(3*x)*K[3] + 12*E^(2*x)*K[4] - 
 
     (49*E^(3*x)*K[5])/2 - 15*E^(3*x)*x*K[5]}, 
 
   {2*E^x*K[1] + 6*E^(2*x)*K[4] + 15*E^(3*x)*K[5]}, 
 
   {-6*E^(2*x)*K[4]}}]
;[o]
    x           2 x         3 x            2 x
-3 E  K[1] - 2 E    K[2] - E    K[3] - 16 E    K[4] - 
 
                       3 x
      2 x          11 E    K[5]       3 x
  12 E    x K[4] - ------------ + 15 E    x K[5]
                        2

   x         2 x         3 x           2 x
5 E  K[1] + E    K[2] - E    K[3] - 7 E    K[4] + 
 
                      3 x
     2 x          49 E    K[5]       3 x
  6 E    x K[4] + ------------ + 15 E    x K[5]
                       2

                                            3 x
    x         3 x            2 x        49 E    K[5]
-5 E  K[1] + E    K[3] + 12 E    K[4] - ------------ - 
                                             2
 
      3 x
  15 E    x K[5]

   x           2 x            3 x
2 E  K[1] + 6 E    K[4] + 15 E    K[5]

    2 x
-6 E    K[4]
:[font = input; initialization; preserveAspect; startGroup]
*)
D[y,x]-M.y//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0}, {0}, {0}, {0}, {0}}]
;[o]
0

0

0

0

0
:[font = input; initialization; preserveAspect]
*)
Exercise 5
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M={{-205,-409,-613,-817,-1021,-1225},   \
   {494,986,1479,1972,2465,2958},       \
   {-662,-1323,-1986,-2648,-3310,-3972},\
   {539,1078,1618,2156,2695,3234},      \
   {-272,-544,-816,-1087,-1360,-1632},  \
   {70,140,210,280,351,421}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-205, -409, -613, -817, -1021, -1225}, 
 
   {494, 986, 1479, 1972, 2465, 2958}, 
 
   {-662, -1323, -1986, -2648, -3310, -3972}, 
 
   {539, 1078, 1618, 2156, 2695, 3234}, 
 
   {-272, -544, -816, -1087, -1360, -1632}, 
 
   {70, 140, 210, 280, 351, 421}}]
;[o]
-205    -409    -613    -817    -1021   -1225

494     986     1479    1972    2465    2958

-662    -1323   -1986   -2648   -3310   -3972

539     1078    1618    2156    2695    3234

-272    -544    -816    -1087   -1360   -1632

70      140     210     280     351     421
:[font = input; initialization; preserveAspect; startGroup]
*)
EV=Eigensystem[M][[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-168, 325, -337, 202, -70, 12}, 
 
   {0, 0, 0, 0, 0, 0}, {-93, 200, -231, 154, -59, 11}, 
 
   {0, 0, 0, 0, 0, 0}, {-64, 143, -173, 122, -50, 10}, 
 
   {0, 0, 0, 0, 0, 0}}]
;[o]
-168   325    -337   202    -70    12

0      0      0      0      0      0

-93    200    -231   154    -59    11

0      0      0      0      0      0

-64    143    -173   122    -50    10

0      0      0      0      0      0
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[EV]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{3590, 3937, 2137, 0, 0, 3647}, 
 
   {-9717, -11218, -6732, 0, 3647, 0}, 
 
   {7824, 10100, 8026, 3647, 0, 0}}]
;[o]
3590     3937     2137     0        0        3647

-9717    -11218   -6732    0        3647     0

7824     10100    8026     3647     0        0
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Join[{EV[[1]]},{EV[[3]]},{EV[[5]]},%]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-168, -93, -64, 3590, -9717, 7824}, 
 
   {325, 200, 143, 3937, -11218, 10100}, 
 
   {-337, -231, -173, 2137, -6732, 8026}, 
 
   {202, 154, 122, 0, 0, 3647}, {-70, -59, -50, 0, 3647, 0}, 
 
   {12, 11, 10, 3647, 0, 0}}]
;[o]
-168     -93      -64      3590     -9717    7824

325      200      143      3937     -11218   10100

-337     -231     -173     2137     -6732    8026

202      154      122      0        0        3647

-70      -59      -50      0        3647     0

12       11       10       3647     0        0
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].M.P; T//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, 123731863945233/1309474756, 
 
    -26854095075623/327368689, 51979335296230/327368689}, 
 
   {0, 2, 0, -478078118052553/654737378, 
 
    206378534111209/327368689, -401302318801220/327368689}, 
 
   {0, 0, 3, 1234440253205609/1309474756, 
 
    -265886302282338/327368689, 517934756801526/327368689}, 
 
   {0, 0, 0, 84391125848213/1193913608783, 
 
    -69607679841880/1193913608783, 
 
    139481166102100/1193913608783}, 
 
   {0, 0, 0, -13607792540137/341118173938, 
 
    5698346761619/170559086969, -11244977904331/170559086969}
 
    , {0, 0, 0, -70981009009520/1193913608783, 
 
    58683893502254/1193913608783, 
 
    -117116071526848/1193913608783}}]
;[o]
            123731863945233        26854095075623
            ---------------      -(--------------)
1   0   0     1309474756             327368689
 
   51979335296230
   --------------
     327368689

              478078118052553    206378534111209
            -(---------------)   ---------------
0   2   0        654737378          327368689
 
     401302318801220
   -(---------------)
        327368689

            1234440253205609       265886302282338
            ----------------     -(---------------)
0   0   3      1309474756             327368689
 
   517934756801526
   ---------------
      327368689

            84391125848213         69607679841880
            --------------       -(--------------)
0   0   0   1193913608783          1193913608783
 
   139481166102100
   ---------------
    1193913608783

              13607792540137     5698346761619
            -(--------------)    -------------
0   0   0      341118173938      170559086969
 
     11244977904331
   -(--------------)
      170559086969

              70981009009520     58683893502254
            -(--------------)    --------------
0   0   0     1193913608783      1193913608783
 
     117116071526848
   -(---------------)
      1193913608783
:[font = input; initialization; preserveAspect; startGroup]
*)
B=Array[T[[3+#1,3+#2]]&,{3,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{84391125848213/1193913608783, 
 
    -69607679841880/1193913608783, 
 
    139481166102100/1193913608783}, 
 
   {-13607792540137/341118173938, 
 
    5698346761619/170559086969, -11244977904331/170559086969}
 
    , {-70981009009520/1193913608783, 
 
    58683893502254/1193913608783, 
 
    -117116071526848/1193913608783}}]
;[o]
84391125848213         69607679841880     139481166102100
--------------       -(--------------)    ---------------
1193913608783          1193913608783       1193913608783

  13607792540137     5698346761619          11244977904331
-(--------------)    -------------        -(--------------)
   341118173938      170559086969            170559086969

  70981009009520     58683893502254         117116071526848
-(--------------)    --------------       -(---------------)
  1193913608783      1193913608783           1193913608783
:[font = input; initialization; preserveAspect; startGroup]
*)
EVB=Eigensystem[B][[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-199614164/233563639, 458868527/467127278, 1}, 
 
   {-50535780/48811633, 38274553/48811633, 1}, 
 
   {-14389730/13219447, 19567917/26438894, 1}}]
;[o]
  199614164    458868527
-(---------)   ---------
  233563639    467127278      1

  50535780     38274553
-(--------)    --------
  48811633     48811633       1

  14389730     19567917
-(--------)    --------
  13219447     26438894       1
:[font = input; initialization; preserveAspect; startGroup]
*)
PB=Transpose[EVB]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-199614164/233563639, -50535780/48811633, 
 
    -14389730/13219447}, 
 
   {458868527/467127278, 38274553/48811633, 
 
    19567917/26438894}, {1, 1, 1}}]
;[o]
  199614164      50535780       14389730
-(---------)   -(--------)    -(--------)
  233563639      48811633       13219447

458868527      38274553       19567917
---------      --------       --------
467127278      48811633       26438894



1              1              1
:[font = input; initialization; preserveAspect; startGroup]
*)
Table[If[i<4 || j<4,IdentityMatrix[6][[i,j]],PB[[i-3,j-3]]], \
                                                {i,6},{j,6}];
%//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, 
 
   {0, 0, 1, 0, 0, 0}, {0, 0, 0, -199614164/233563639, 
 
    -50535780/48811633, -14389730/13219447}, 
 
   {0, 0, 0, 458868527/467127278, 38274553/48811633, 
 
    19567917/26438894}, {0, 0, 0, 1, 1, 1}}]
;[o]
1   0   0   0              0             0

0   1   0   0              0             0

0   0   1   0              0             0

              199614164      50535780      14389730
            -(---------)   -(--------)   -(--------)
0   0   0     233563639      48811633      13219447

            458868527      38274553      19567917
            ---------      --------      --------
0   0   0   467127278      48811633      26438894

0   0   0   1              1             1
:[font = input; initialization; preserveAspect; startGroup]
*)
P=P.%%; P//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{-168, -93, -64, -2237251351307/467127278, 
 
    -171435065109/48811633, -86601804233/26438894}, 
 
   {325, 200, 143, -1000681777711/233563639, 
 
    -135325808114/48811633, -1134220647/455843}, 
 
   {-337, -231, -173, -96545163736/233563639, 
 
    26102913802/48811633, 9482819990/13219447}, 
 
   {202, 154, 122, 3647, 3647, 3647}, 
 
   {-70, -59, -50, 1673493517969/467127278, 
 
    139587294791/48811633, 71364193299/26438894}, 
 
   {12, 11, 10, -727992856108/233563639, 
 
    -184303989660/48811633, -52479345310/13219447}}]
;[o]
                       2237251351307      171435065109
                     -(-------------)   -(------------)
-168   -93    -64        467127278          48811633
 
        86601804233
      -(-----------)
         26438894

                       1000681777711      135325808114
                     -(-------------)   -(------------)
325    200    143        233563639          48811633
 
        1134220647
      -(----------)
          455843

                       96545163736      26102913802
                     -(-----------)     -----------
-337   -231   -173      233563639        48811633
 
      9482819990
      ----------
       13219447

202    154    122    3647               3647
 
      3647

                     1673493517969      139587294791
                     -------------      ------------
-70    -59    -50      467127278          48811633
 
      71364193299
      -----------
       26438894

                       727992856108       184303989660
                     -(------------)    -(------------)
12     11     10        233563639           48811633
 
        52479345310
      -(-----------)
         13219447
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].M.P; T//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, -1193913608783/467127278, 
 
    -12654236602/3754741, -63287397572/13219447}, 
 
   {0, 2, 0, 8163633441075/467127278, 
 
    1193913608783/48811633, 940296385567/26438894}, 
 
   {0, 0, 3, -4996327439362/233563639, 
 
    -1500718862349/48811633, -1193913608783/26438894}, 
 
   {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 2, 0}, 
 
   {0, 0, 0, 0, 0, 3}}]
;[o]
              1193913608783      12654236602
            -(-------------)   -(-----------)
1   0   0       467127278          3754741
 
     63287397572
   -(-----------)
      13219447

            8163633441075      1193913608783
            -------------      -------------
0   2   0     467127278          48811633
 
   940296385567
   ------------
     26438894

              4996327439362      1500718862349
            -(-------------)   -(-------------)
0   0   3       233563639          48811633
 
     1193913608783
   -(-------------)
       26438894

0   0   0   1                  0
 
   0

0   0   0   0                  2
 
   0

0   0   0   0                  0
 
   3
:[font = input; initialization; preserveAspect; startGroup]
*)
z=Array[0&,6];
Do[z[[7-i]]=Z[7-i][x]/.DSolve[Z[7-i]'[x]==T[[7-i,7-i]]Z[7-i][x] \
                            +Sum[T[[7-i,j]]z[[j,1]],{j,8-i,6}], \
                            Z[7-i][x],x]/.{C[1]->K[7-i]},{i,6}]
z//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{E^x*K[1] - (1193913608783*E^x*x*K[4])/
 
      467127278 - (12654236602*E^(2*x)*K[5])/3754741 - 
 
     (31643698786*E^(3*x)*K[6])/13219447}, 
 
   {E^(2*x)*K[2] - (8163633441075*E^x*K[4])/467127278 + 
 
     (1193913608783*E^(2*x)*x*K[5])/48811633 + 
 
     (940296385567*E^(3*x)*K[6])/26438894}, 
 
   {E^(3*x)*K[3] + (2498163719681*E^x*K[4])/233563639 + 
 
     (1500718862349*E^(2*x)*K[5])/48811633 - 
 
     (1193913608783*E^(3*x)*x*K[6])/26438894}, {E^x*K[4]}, 
 
   {E^(2*x)*K[5]}, {E^(3*x)*K[6]}}]
;[o]
                         x                       2 x
 x        1193913608783 E  x K[4]   12654236602 E    K[5]
E  K[1] - ----------------------- - --------------------- - 
                 467127278                 3754741
 
               3 x
  31643698786 E    K[6]
  ---------------------
        13219447

                           x
 2 x        8163633441075 E  K[4]
E    K[2] - --------------------- + 
                  467127278
 
                 2 x                        3 x
  1193913608783 E    x K[5]   940296385567 E    K[6]
  ------------------------- + ----------------------
          48811633                   26438894

                           x
 3 x        2498163719681 E  K[4]
E    K[3] + --------------------- + 
                  233563639
 
                 2 x                       3 x
  1500718862349 E    K[5]   1193913608783 E    x K[6]
  ----------------------- - -------------------------
         48811633                   26438894

 x
E  K[4]

 2 x
E    K[5]

 3 x
E    K[6]
:[font = input; initialization; preserveAspect; startGroup]
*)
y=P.z//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-168*E^x*K[1] - 93*E^(2*x)*K[2] - 
 
     64*E^(3*x)*K[3] + (218607851274750*E^x*K[4])/
 
      233563639 + (100288743137772*E^x*x*K[4])/233563639 - 
 
     (68580589516677*E^(2*x)*K[5])/48811633 - 
 
     (111033965616819*E^(2*x)*x*K[5])/48811633 - 
 
     (38450941434934*E^(3*x)*K[6])/13219447 + 
 
     (38205235481056*E^(3*x)*x*K[6])/13219447}, 
 
   {325*E^x*K[1] + 200*E^(2*x)*K[2] + 143*E^(3*x)*K[3] - 
 
     (460126613970828*E^x*K[4])/233563639 - 
 
     (388021922854475*E^x*x*K[4])/467127278 + 
 
     (161003321864343*E^(2*x)*K[5])/48811633 + 
 
     (238782721756600*E^(2*x)*x*K[5])/48811633 + 
 
     (83712544052487*E^(3*x)*K[6])/13219447 - 
 
     (170729646055969*E^(3*x)*x*K[6])/26438894}, 
 
   {-337*E^x*K[1] - 231*E^(2*x)*K[2] - 173*E^(3*x)*K[3] + 
 
     (1021241587551227*E^x*K[4])/467127278 + 
 
     (402348886159871*E^x*x*K[4])/467127278 - 
 
     (204160049719213*E^(2*x)*K[5])/48811633 - 
 
     (275794043628873*E^(2*x)*x*K[5])/48811633 - 
 
     (4554922010331*E^(3*x)*K[6])/614858 + 
 
     (206547054319459*E^(3*x)*x*K[6])/26438894}, 
 
   {202*E^x*K[1] + 154*E^(2*x)*K[2] + 122*E^(3*x)*K[3] - 
 
     (322971994570260*E^x*K[4])/233563639 - 
 
     (120585274487083*E^x*x*K[4])/233563639 + 
 
     (150035691915277*E^(2*x)*K[5])/48811633 + 
 
     (183862695752582*E^(2*x)*x*K[5])/48811633 + 
 
     (66059005857096*E^(3*x)*K[6])/13219447 - 
 
     (72828730135763*E^(3*x)*x*K[6])/13219447}, 
 
   {-70*E^x*K[1] - 59*E^(2*x)*K[2] - 50*E^(3*x)*K[3] + 
 
     (116755747286647*E^x*K[4])/233563639 + 
 
     (41786976307405*E^x*x*K[4])/233563639 - 
 
     (63381000514839*E^(2*x)*K[5])/48811633 - 
 
     (70440902918197*E^(2*x)*x*K[5])/48811633 - 
 
     (25488002362557*E^(3*x)*K[6])/13219447 + 
 
     (29847840219575*E^(3*x)*x*K[6])/13219447}, 
 
   {12*E^x*K[1] + 11*E^(2*x)*K[2] + 10*E^(3*x)*K[3] - 
 
     (41292679170421*E^x*K[4])/467127278 - 
 
     (7163481652698*E^x*x*K[4])/233563639 + 
 
     (988371055686*E^(2*x)*K[5])/3754741 + 
 
     (13133049696613*E^(2*x)*x*K[5])/48811633 + 
 
     (9478852779753*E^(3*x)*K[6])/26438894 - 
 
     (5969568043915*E^(3*x)*x*K[6])/13219447}}]
;[o]
      x            2 x            3 x
-168 E  K[1] - 93 E    K[2] - 64 E    K[3] + 
 
                   x                         x
  218607851274750 E  K[4]   100288743137772 E  x K[4]
  ----------------------- + ------------------------- - 
         233563639                  233563639
 
                  2 x                         2 x
  68580589516677 E    K[5]   111033965616819 E    x K[5]
  ------------------------ - --------------------------- - 
          48811633                    48811633
 
                  3 x                        3 x
  38450941434934 E    K[6]   38205235481056 E    x K[6]
  ------------------------ + --------------------------
          13219447                    13219447

     x             2 x             3 x
325 E  K[1] + 200 E    K[2] + 143 E    K[3] - 
 
                   x                         x
  460126613970828 E  K[4]   388021922854475 E  x K[4]
  ----------------------- - ------------------------- + 
         233563639                  467127278
 
                   2 x                         2 x
  161003321864343 E    K[5]   238782721756600 E    x K[5]
  ------------------------- + --------------------------- + 
          48811633                     48811633
 
                  3 x                         3 x
  83712544052487 E    K[6]   170729646055969 E    x K[6]
  ------------------------ - ---------------------------
          13219447                    26438894

      x             2 x             3 x
-337 E  K[1] - 231 E    K[2] - 173 E    K[3] + 
 
                    x                         x
  1021241587551227 E  K[4]   402348886159871 E  x K[4]
  ------------------------ + ------------------------- - 
         467127278                   467127278
 
                   2 x                         2 x
  204160049719213 E    K[5]   275794043628873 E    x K[5]
  ------------------------- - --------------------------- - 
          48811633                     48811633
 
                 3 x                         3 x
  4554922010331 E    K[6]   206547054319459 E    x K[6]
  ----------------------- + ---------------------------
          614858                     26438894

     x             2 x             3 x
202 E  K[1] + 154 E    K[2] + 122 E    K[3] - 
 
                   x                         x
  322971994570260 E  K[4]   120585274487083 E  x K[4]
  ----------------------- - ------------------------- + 
         233563639                  233563639
 
                   2 x                         2 x
  150035691915277 E    K[5]   183862695752582 E    x K[5]
  ------------------------- + --------------------------- + 
          48811633                     48811633
 
                  3 x                        3 x
  66059005857096 E    K[6]   72828730135763 E    x K[6]
  ------------------------ - --------------------------
          13219447                    13219447

     x            2 x            3 x
-70 E  K[1] - 59 E    K[2] - 50 E    K[3] + 
 
                   x                        x
  116755747286647 E  K[4]   41786976307405 E  x K[4]
  ----------------------- + ------------------------ - 
         233563639                 233563639
 
                  2 x                        2 x
  63381000514839 E    K[5]   70440902918197 E    x K[5]
  ------------------------ - -------------------------- - 
          48811633                    48811633
 
                  3 x                        3 x
  25488002362557 E    K[6]   29847840219575 E    x K[6]
  ------------------------ + --------------------------
          13219447                    13219447

    x            2 x            3 x
12 E  K[1] + 11 E    K[2] + 10 E    K[3] - 
 
                  x                       x
  41292679170421 E  K[4]   7163481652698 E  x K[4]
  ---------------------- - ----------------------- + 
        467127278                 233563639
 
                2 x                        2 x
  988371055686 E    K[5]   13133049696613 E    x K[5]
  ---------------------- + -------------------------- + 
         3754741                    48811633
 
                 3 x                       3 x
  9478852779753 E    K[6]   5969568043915 E    x K[6]
  ----------------------- - -------------------------
         26438894                   13219447
:[font = input; initialization; preserveAspect; startGroup]
*)
D[y,x]-M.y//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0}, {0}, {0}, {0}, {0}, {0}}]
;[o]
0

0

0

0

0

0
:[font = input; initialization; preserveAspect]
*)
Problems on Differential Equations
(*
:[font = input; initialization; preserveAspect]
*)
Problem 1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M={{0,1},  \
   {-2,-1}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1}, {-2, -1}}]
;[o]
0    1

-2   -1
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Eigensystem[M][[2]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{I/4*(I + 7^(1/2)), -I/4*(-I + 7^(1/2))}, 
 
   {1, 1}}]
;[o]
I                   -I
- (I + Sqrt[7])     -- (-I + Sqrt[7])
4                   4



1                   1
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].M.P//Together
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-I/2*(-I + 7^(1/2)), 0}, 
 
   {0, I/2*(I + 7^(1/2))}}]
;[o]
-I
-- (-I + Sqrt[7])
2                   0

                    I
                    - (I + Sqrt[7])
0                   2
:[font = input; initialization; preserveAspect; startGroup]
*)
k={0,E^x}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{0, E^x}
;[o]
     x
{0, E }
:[font = input; initialization; preserveAspect; startGroup]
*)
IPk=Inverse[P].k//Together
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{((-I + 7^(1/2))*E^x)/(2*7^(1/2)), 
 
  ((I + 7^(1/2))*E^x)/(2*7^(1/2))}
;[o]
                 x                 x
 (-I + Sqrt[7]) E   (I + Sqrt[7]) E
{-----------------, ----------------}
     2 Sqrt[7]         2 Sqrt[7]
:[font = input; initialization; preserveAspect; startGroup]
*)
z=Table[Z[i][x]/.DSolve[Z[i]'[x]==T[[i,i]]Z[i][x]+IPk[[i]], \
                        Z[i][x],x]/.{C[1]->K[i]},{i,2}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-((7*I + 7^(1/2))*E^x)/(7*(-3*I + 7^(1/2))) + 
 
     E^((-1/2 - I/2*7^(1/2))*x)*K[1]}, 
 
   {-((-7*I + 7^(1/2))*E^x)/(7*(3*I + 7^(1/2))) + 
 
     E^((-1/2 + I/2*7^(1/2))*x)*K[2]}}]
;[o]
                   x
-((7 I + Sqrt[7]) E )    (-(1/2) - I/2 Sqrt[7]) x
--------------------- + E                         K[1]
 7 (-3 I + Sqrt[7])

                    x
-((-7 I + Sqrt[7]) E )    (-(1/2) + I/2 Sqrt[7]) x
---------------------- + E                         K[2]
  7 (3 I + Sqrt[7])
:[font = input; initialization; preserveAspect; startGroup]
*)
u=P.z//Together//Simplify
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(E^x - E^((-1/2 - I/2*7^(1/2))*x)*K[1] + 
 
       I*7^(1/2)*E^((-1/2 - I/2*7^(1/2))*x)*K[1] - 
 
       E^(I/2*(I + 7^(1/2))*x)*K[2] - 
 
       I*7^(1/2)*E^(I/2*(I + 7^(1/2))*x)*K[2])/4}, 
 
   {E^x/4 + E^((-1/2 - I/2*7^(1/2))*x)*K[1] + 
 
     E^(I/2*(I + 7^(1/2))*x)*K[2]}}]
;[o]
  x    (-(1/2) - I/2 Sqrt[7]) x
(E  - E                         K[1] + 
 
               (-(1/2) - I/2 Sqrt[7]) x
    I Sqrt[7] E                         K[1] - 
 
     I/2 (I + Sqrt[7]) x
    E                    K[2] - 
 
               I/2 (I + Sqrt[7]) x
    I Sqrt[7] E                    K[2]) / 4

 x
E     (-(1/2) - I/2 Sqrt[7]) x
-- + E                         K[1] + 
4
 
   I/2 (I + Sqrt[7]) x
  E                    K[2]
:[font = input; initialization; preserveAspect; startGroup]
*)
yRe=Re[u[[1]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{Re[(E^x - E^((-1/2 - I/2*7^(1/2))*x)*K[1] + 
 
      I*7^(1/2)*E^((-1/2 - I/2*7^(1/2))*x)*K[1] - 
 
      E^(I/2*(I + 7^(1/2))*x)*K[2] - 
 
      I*7^(1/2)*E^(I/2*(I + 7^(1/2))*x)*K[2])/4]}
;[o]
      x    (-(1/2) - I/2 Sqrt[7]) x
{Re[(E  - E                         K[1] + 
 
                 (-(1/2) - I/2 Sqrt[7]) x
      I Sqrt[7] E                         K[1] - 
 
       I/2 (I + Sqrt[7]) x
      E                    K[2] - 
 
                 I/2 (I + Sqrt[7]) x
      I Sqrt[7] E                    K[2]) / 4]}
:[font = input; initialization; preserveAspect]
*)
Mathematica does not know if the constants are real
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
yRe=(E^x-E^(-x/2)Cos[Sqrt[7]x/2](K[1]-K[2])+       \
     Sqrt[7]E^(-x/2)Sin[Sqrt[7]x/2](K[1]+K[2]))/4
(*
:[font = output; output; inactive; preserveAspect; endGroup]
(E^x - (Cos[(7^(1/2)*x)/2]*(K[1] - K[2]))/E^(x/2) + 
 
    (7^(1/2)*(K[1] + K[2])*Sin[(7^(1/2)*x)/2])/E^(x/2))/4
;[o]
          Sqrt[7] x
      Cos[---------] (K[1] - K[2])
  x           2
(E  - ---------------------------- + 
                   x/2
                  E
 
                              Sqrt[7] x
    Sqrt[7] (K[1] + K[2]) Sin[---------]
                                  2
    ------------------------------------) / 4
                     x/2
                    E
:[font = input; initialization; preserveAspect]
*)
Solution check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
D[D[yRe,x],x]+D[yRe,x]+2yRe//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
E^x
;[o]
 x
E
:[font = input; initialization; preserveAspect]
*)
Problem 2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M={{1/75,-1/150},     \
   {1/75,-1/300}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1/75, -1/150}, {1/75, -1/300}}]
;[o]
1           1
--       -(---)
75         150

1           1
--       -(---)
75         300
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Eigensystem[M][[2]]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-I/8*(5*I + 7^(1/2)), I/8*(-5*I + 7^(1/2))}, 
 
   {1, 1}}]
;[o]
-I                   I
-- (5 I + Sqrt[7])   - (-5 I + Sqrt[7])
8                    8



1                    1
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].M.P//Together
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(3 - I*7^(1/2))/600, 0}, 
 
   {0, (3 + I*7^(1/2))/600}}]
;[o]
3 - I Sqrt[7]
-------------
     600        0

                3 + I Sqrt[7]
                -------------
0                    600
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[K]
z=Table[Z[i][t]/.DSolve[Z[i]'[t]==T[[i,i]]Z[i][t],Z[i][t],t] \
                                        /.{C[1]->K[i]},{i,2}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{E^((1/200 - I/600*7^(1/2))*t)*K[1]}, 
 
   {E^((1/200 + I/600*7^(1/2))*t)*K[2]}}]
;[o]
 (1/200 - I/600 Sqrt[7]) t
E                          K[1]

 (1/200 + I/600 Sqrt[7]) t
E                          K[2]
:[font = input; initialization; preserveAspect; startGroup]
*)
r=(P.z)[[1]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-I/8*(5*I + 7^(1/2))*E^((1/200 - I/600*7^(1/2))*t)*K[1] + 
 
   I/8*(-5*I + 7^(1/2))*E^((1/200 + I/600*7^(1/2))*t)*K[2]}
;[o]
 -I                  (1/200 - I/600 Sqrt[7]) t
{-- (5 I + Sqrt[7]) E                          K[1] + 
 8
 
   I                   (1/200 + I/600 Sqrt[7]) t
   - (-5 I + Sqrt[7]) E                          K[2]}
   8
:[font = input; initialization; preserveAspect; startGroup]
*)
f=(P.z)[[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{E^((1/200 - I/600*7^(1/2))*t)*K[1] + 
 
   E^((1/200 + I/600*7^(1/2))*t)*K[2]}
;[o]
  (1/200 - I/600 Sqrt[7]) t
{E                          K[1] + 
 
    (1/200 + I/600 Sqrt[7]) t
   E                          K[2]}
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Solve[(r/.{t->0})-1000==0,K[2]]//Expand;
Solve[(f/.{t->0}/.%)-1000==0,K[1]]//Expand;
Re[r/.{t->{10,20,100}}/.%%/.%//N]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{{1067.764700186698003, 1137.667110988617783, 
 
     1756.977404441146643}}}}
;[o]
{{{{1067.76, 1137.67, 1756.98}}}}
:[font = input; initialization; preserveAspect; startGroup]
*)
Re[f/.{t->{10,20,100}}/.%%%/.%%//N]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{{1102.795714846362742, 1211.249700731157621, 
 
     2288.913256528842839}}}}
;[o]
{{{{1102.8, 1211.25, 2288.91}}}}
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Solve[(r/.{t->0})-2000==0,K[2]]//Expand;
Solve[(f/.{t->0}/.%)-1000==0,K[1]]//Expand;
Re[r/.{t->{10,20,100}}/.%%/.%//N]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{{2205.591429692725483, 2422.499401462315243, 
 
     4577.826513057685679}}}}
;[o]
{{{{2205.59, 2422.5, 4577.83}}}}
:[font = input; initialization; preserveAspect; startGroup]
*)
Re[f/.{t->{10,20,100}}/.%%%/.%%//N]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{{1242.919773485021695, 1505.580059701316973, 
 
     4416.656664879627626}}}}
;[o]
{{{{1242.92, 1505.58, 4416.66}}}}
:[font = input; initialization; preserveAspect]
*)
Problem 3
(*
:[font = input; initialization; preserveAspect]
*)
Since both w1/g and w2/g are equal to 1, we will just
skip writing them.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
k1=12; k2=10; k3=12;
M={{0,1,0,0},      \
   {-k1-k2,0,k2,0},\
   {0,0,0,1},      \
   {k2,0,-k2-k3,0}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1, 0, 0}, {-22, 0, 10, 0}, {0, 0, 0, 1}, 
 
   {10, 0, -22, 0}}]
;[o]
0     1     0     0

-22   0     10    0

0     0     0     1

10    0     -22   0
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Eigensystem[M][[2]]]//Together
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-I/4/2^(1/2), I/4/2^(1/2), I/2/3^(1/2), 
 
    -I/2/3^(1/2)}, {-1, -1, 1, 1}, 
 
   {I/4/2^(1/2), -I/4/2^(1/2), I/2/3^(1/2), -I/2/3^(1/2)}, 
 
   {1, 1, 1, 1}}]
;[o]
  -I         I         I        -I
  --         -         -        --
  4          4         2        2
-------   -------   -------   -------
Sqrt[2]   Sqrt[2]   Sqrt[3]   Sqrt[3]





-1        -1        1         1

   I        -I         I        -I
   -        --         -        --
   4        4          2        2
-------   -------   -------   -------
Sqrt[2]   Sqrt[2]   Sqrt[3]   Sqrt[3]





1         1         1         1
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].M.P
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-4*I*2^(1/2), 0, 0, 0}, {0, 4*I*2^(1/2), 0, 0}, 
 
   {0, 0, -2*I*3^(1/2), 0}, {0, 0, 0, 2*I*3^(1/2)}}]
;[o]
-4 I Sqrt[2]   0              0              0

0              4 I Sqrt[2]    0              0

0              0              -2 I Sqrt[3]   0

0              0              0              2 I Sqrt[3]
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[K]
z=Table[Z[i][t]/.DSolve[Z[i]'[t]==T[[i,i]]Z[i][t],Z[i][t],t] \
                                        /.{C[1]->K[i]},{i,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{E^(-4*I*2^(1/2)*t)*K[1]}, 
 
   {E^(4*I*2^(1/2)*t)*K[2]}, {E^(-2*I*3^(1/2)*t)*K[3]}, 
 
   {E^(2*I*3^(1/2)*t)*K[4]}}]
;[o]
 -4 I Sqrt[2] t
E               K[1]

 4 I Sqrt[2] t
E              K[2]

 -2 I Sqrt[3] t
E               K[3]

 2 I Sqrt[3] t
E              K[4]
:[font = input; initialization; preserveAspect; startGroup]
*)
u=P.z
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(-I/4*E^(-4*I*2^(1/2)*t)*K[1])/2^(1/2) + 
 
     (I/4*E^(4*I*2^(1/2)*t)*K[2])/2^(1/2) + 
 
     (I/2*E^(-2*I*3^(1/2)*t)*K[3])/3^(1/2) - 
 
     (I/2*E^(2*I*3^(1/2)*t)*K[4])/3^(1/2)}, 
 
   {-(E^(-4*I*2^(1/2)*t)*K[1]) - E^(4*I*2^(1/2)*t)*K[2] + 
 
     E^(-2*I*3^(1/2)*t)*K[3] + E^(2*I*3^(1/2)*t)*K[4]}, 
 
   {(I/4*E^(-4*I*2^(1/2)*t)*K[1])/2^(1/2) - 
 
     (I/4*E^(4*I*2^(1/2)*t)*K[2])/2^(1/2) + 
 
     (I/2*E^(-2*I*3^(1/2)*t)*K[3])/3^(1/2) - 
 
     (I/2*E^(2*I*3^(1/2)*t)*K[4])/3^(1/2)}, 
 
   {E^(-4*I*2^(1/2)*t)*K[1] + E^(4*I*2^(1/2)*t)*K[2] + 
 
     E^(-2*I*3^(1/2)*t)*K[3] + E^(2*I*3^(1/2)*t)*K[4]}}]
;[o]
-I  -4 I Sqrt[2] t        I  4 I Sqrt[2] t
-- E               K[1]   - E              K[2]
4                         4
----------------------- + --------------------- + 
        Sqrt[2]                  Sqrt[2]
 
  I  -2 I Sqrt[3] t        I  2 I Sqrt[3] t
  - E               K[3]   - E              K[4]
  2                        2
  ---------------------- - ---------------------
         Sqrt[3]                  Sqrt[3]

   -4 I Sqrt[2] t          4 I Sqrt[2] t
-(E               K[1]) - E              K[2] + 
 
   -2 I Sqrt[3] t         2 I Sqrt[3] t
  E               K[3] + E              K[4]

I  -4 I Sqrt[2] t        I  4 I Sqrt[2] t
- E               K[1]   - E              K[2]
4                        4
---------------------- - --------------------- + 
       Sqrt[2]                  Sqrt[2]
 
  I  -2 I Sqrt[3] t        I  2 I Sqrt[3] t
  - E               K[3]   - E              K[4]
  2                        2
  ---------------------- - ---------------------
         Sqrt[3]                  Sqrt[3]

 -4 I Sqrt[2] t         4 I Sqrt[2] t
E               K[1] + E              K[2] + 
 
   -2 I Sqrt[3] t         2 I Sqrt[3] t
  E               K[3] + E              K[4]
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[y]
Solve[(u[[1]]/.{t->0})-10==0,K[1]]//Expand;
Solve[(u[[2]]/.{t->0}/.%)==0,K[2]]//Expand;
Solve[(u[[3]]/.{t->0}/.%%/.%)-12==0,K[3]]//Expand;
Solve[(u[[4]]/.{t->0}/.%%%/.%%/.%)==0,K[4]]//Expand;
y[1]=u[[1]]/.%%%%/.%%%/.%%/.%//ComplexExpand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{{{-Cos[4*2^(1/2)*t] + 11*Cos[2*3^(1/2)*t]}}}}}
;[o]
{{{{{-Cos[4 Sqrt[2] t] + 11 Cos[2 Sqrt[3] t]}}}}}
:[font = input; initialization; preserveAspect; startGroup]
*)
y[2]=u[[3]]/.%%%%%/.%%%%/.%%%/.%%//ComplexExpand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{{{Cos[4*2^(1/2)*t] + 11*Cos[2*3^(1/2)*t]}}}}}
;[o]
{{{{{Cos[4 Sqrt[2] t] + 11 Cos[2 Sqrt[3] t]}}}}}
:[font = input; initialization; preserveAspect]
*)
Exercises on the Matrix Exponential Function
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
J3={{2,1,0},  \
    {0,2,1},  \
    {0,0,2}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 1, 0}, {0, 2, 1}, {0, 0, 2}}]
;[o]
2   1   0

0   2   1

0   0   2
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixExp[J3 t]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{E^(2*t), E^(2*t)*t, (E^(2*t)*t^2)/2}, 
 
   {0, E^(2*t), E^(2*t)*t}, {0, 0, E^(2*t)}}]
;[o]
                     2 t  2
                    E    t
 2 t       2 t      -------
E         E    t       2



           2 t       2 t
0         E         E    t



                     2 t
0         0         E
:[font = input; initialization; preserveAspect]
*)
Exercise 2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[1&,{4,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1, 1, 1}, {1, 1, 1, 1}, {1, 1, 1, 1}, 
 
   {1, 1, 1, 1}}]
;[o]
1   1   1   1

1   1   1   1

1   1   1   1

1   1   1   1
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixExp[A t]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{3/4 + E^(4*t)/4, -1/4 + E^(4*t)/4, 
 
    -1/4 + E^(4*t)/4, -1/4 + E^(4*t)/4}, 
 
   {-1/4 + E^(4*t)/4, 3/4 + E^(4*t)/4, -1/4 + E^(4*t)/4, 
 
    -1/4 + E^(4*t)/4}, {-1/4 + E^(4*t)/4, -1/4 + E^(4*t)/4, 
 
    3/4 + E^(4*t)/4, -1/4 + E^(4*t)/4}, 
 
   {-1/4 + E^(4*t)/4, -1/4 + E^(4*t)/4, -1/4 + E^(4*t)/4, 
 
    3/4 + E^(4*t)/4}}]
;[o]
     4 t              4 t           4 t           4 t
3   E           1    E        1    E        1    E
- + ----      -(-) + ----   -(-) + ----   -(-) + ----
4    4          4     4       4     4       4     4

        4 t        4 t              4 t           4 t
  1    E      3   E           1    E        1    E
-(-) + ----   - + ----      -(-) + ----   -(-) + ----
  4     4     4    4          4     4       4     4

        4 t           4 t        4 t              4 t
  1    E        1    E      3   E           1    E
-(-) + ----   -(-) + ----   - + ----      -(-) + ----
  4     4       4     4     4    4          4     4

        4 t           4 t           4 t        4 t
  1    E        1    E        1    E      3   E
-(-) + ----   -(-) + ----   -(-) + ----   - + ----
  4     4       4     4       4     4     4    4
:[font = input; initialization; preserveAspect]
*)
Exercise 3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[If[i>j,i,0],{i,4},{j,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0}, {2, 0, 0, 0}, {3, 3, 0, 0}, 
 
   {4, 4, 4, 0}}]
;[o]
0   0   0   0

2   0   0   0

3   3   0   0

4   4   4   0
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixExp[A t]//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{1, 0, 0, 0}, 
 
   {-5/3 + (20*t^2 + 24*t^3)/(12*t^2), 1, 0, 0}, 
 
   {2/3 + (4*t + 20*t^2 + 12*t^3)/(4*t) - 
 
     (20*t^2 + 24*t^3)/(12*t^2), -1 + (4*t + 12*t^2)/(4*t), 
 
    1, 0}, {4*t + 10*t^2 + 4*t^3, 4*t + 6*t^2, 4*t, 1}}]
;[o]
1                                         0
 
      0     0

           2       3
  5    20 t  + 24 t
-(-) + -------------
  3            2
           12 t                           1
 




      0     0

              2       3       2       3
2   4 t + 20 t  + 12 t    20 t  + 24 t                   2
- + ------------------- - -------------        4 t + 12 t
3           4 t                   2       -1 + -----------
                              12 t                 4 t
 




      1     0

          2      3                                 2
4 t + 10 t  + 4 t                         4 t + 6 t
 

      4 t   1
:[font = input; initialization; preserveAspect]
*)
Exercise 4
(*
:[font = input; initialization; preserveAspect]
*)
J[n_]:=Table[Which[j>i+1,0,j==i+1,1,j==i,2,j<i,0],{i,n},{j,n}]
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
J[4]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 1, 0, 0}, {0, 2, 1, 0}, {0, 0, 2, 1}, 
 
   {0, 0, 0, 2}}]
;[o]
2   1   0   0

0   2   1   0

0   0   2   1

0   0   0   2
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixExp[J[4]t]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{E^(2*t), E^(2*t)*t, (E^(2*t)*t^2)/2, 
 
    (E^(2*t)*t^3)/6}, {0, E^(2*t), E^(2*t)*t, 
 
    (E^(2*t)*t^2)/2}, {0, 0, E^(2*t), E^(2*t)*t}, 
 
   {0, 0, 0, E^(2*t)}}]
;[o]
                     2 t  2    2 t  3
                    E    t    E    t
 2 t       2 t      -------   -------
E         E    t       2         6

                               2 t  2
                              E    t
           2 t       2 t      -------
0         E         E    t       2



                     2 t       2 t
0         0         E         E    t



                               2 t
0         0         0         E
:[font = input; initialization; preserveAspect; startGroup]
*)
J[5]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 1, 0, 0, 0}, {0, 2, 1, 0, 0}, 
 
   {0, 0, 2, 1, 0}, {0, 0, 0, 2, 1}, {0, 0, 0, 0, 2}}]
;[o]
2   1   0   0   0

0   2   1   0   0

0   0   2   1   0

0   0   0   2   1

0   0   0   0   2
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixExp[J[5]t]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{E^(2*t), E^(2*t)*t, (E^(2*t)*t^2)/2, 
 
    (E^(2*t)*t^3)/6, (E^(2*t)*t^4)/24}, 
 
   {0, E^(2*t), E^(2*t)*t, (E^(2*t)*t^2)/2, 
 
    (E^(2*t)*t^3)/6}, {0, 0, E^(2*t), E^(2*t)*t, 
 
    (E^(2*t)*t^2)/2}, {0, 0, 0, E^(2*t), E^(2*t)*t}, 
 
   {0, 0, 0, 0, E^(2*t)}}]
;[o]
                     2 t  2    2 t  3    2 t  4
                    E    t    E    t    E    t
 2 t       2 t      -------   -------   -------
E         E    t       2         6        24

                               2 t  2    2 t  3
                              E    t    E    t
           2 t       2 t      -------   -------
0         E         E    t       2         6

                                         2 t  2
                                        E    t
                     2 t       2 t      -------
0         0         E         E    t       2



                               2 t       2 t
0         0         0         E         E    t



                                         2 t
0         0         0         0         E
:[font = input; initialization; preserveAspect; startGroup]
*)
J[6]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 1, 0, 0, 0, 0}, {0, 2, 1, 0, 0, 0}, 
 
   {0, 0, 2, 1, 0, 0}, {0, 0, 0, 2, 1, 0}, 
 
   {0, 0, 0, 0, 2, 1}, {0, 0, 0, 0, 0, 2}}]
;[o]
2   1   0   0   0   0

0   2   1   0   0   0

0   0   2   1   0   0

0   0   0   2   1   0

0   0   0   0   2   1

0   0   0   0   0   2
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixExp[J[6]t]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{E^(2*t), E^(2*t)*t, (E^(2*t)*t^2)/2, 
 
    (E^(2*t)*t^3)/6, (E^(2*t)*t^4)/24, (E^(2*t)*t^5)/120}, 
 
   {0, E^(2*t), E^(2*t)*t, (E^(2*t)*t^2)/2, (E^(2*t)*t^3)/6, 
 
    (E^(2*t)*t^4)/24}, {0, 0, E^(2*t), E^(2*t)*t, 
 
    (E^(2*t)*t^2)/2, (E^(2*t)*t^3)/6}, 
 
   {0, 0, 0, E^(2*t), E^(2*t)*t, (E^(2*t)*t^2)/2}, 
 
   {0, 0, 0, 0, E^(2*t), E^(2*t)*t}, 
 
   {0, 0, 0, 0, 0, E^(2*t)}}]
;[o]
                     2 t  2    2 t  3    2 t  4    2 t  5
                    E    t    E    t    E    t    E    t
 2 t       2 t      -------   -------   -------   -------
E         E    t       2         6        24        120

                               2 t  2    2 t  3    2 t  4
                              E    t    E    t    E    t
           2 t       2 t      -------   -------   -------
0         E         E    t       2         6        24

                                         2 t  2    2 t  3
                                        E    t    E    t
                     2 t       2 t      -------   -------
0         0         E         E    t       2         6

                                                   2 t  2
                                                  E    t
                               2 t       2 t      -------
0         0         0         E         E    t       2



                                         2 t       2 t
0         0         0         0         E         E    t



                                                   2 t
0         0         0         0         0         E
:[font = input; initialization; preserveAspect]
*)
Problems on the Matrix Exponential
(*
:[font = input; initialization; preserveAspect]
*)
Problem 1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[If[i<j,i,0],{i,4},{j,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1, 1, 1}, {0, 0, 2, 2}, {0, 0, 0, 3}, 
 
   {0, 0, 0, 0}}]
;[o]
0   1   1   1

0   0   2   2

0   0   0   3

0   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixExp[A t]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, t, t + t^2, t + (5*t^2)/2 + t^3}, 
 
   {0, 1, 2*t, 2*t + 3*t^2}, {0, 0, 1, 3*t}, {0, 0, 0, 1}}]
;[o]
                                                       2
                                                    5 t     3
                                     2          t + ---- + t
1               t               t + t                2



                                                         2
0               1               2 t             2 t + 3 t




0               0               1               3 t




0               0               0               1
:[font = input; initialization; preserveAspect]
*)
Solution check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
D[%,t]-A.%
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, 
 
   {0, 0, 0, 0}}]
;[o]
0   0   0   0

0   0   0   0

0   0   0   0

0   0   0   0
:[font = input; initialization; preserveAspect]
*)
Actually, since every column of E^(At) is a solution of the
differential equation, we get a general solution from:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
y=Transpose[{Sum[Transpose[%%][[i]]K[i],{i,4}]}]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{K[1] + t*K[2] + t*K[3] + t^2*K[3] + t*K[4] + 
 
     (5*t^2*K[4])/2 + t^3*K[4]}, 
 
   {K[2] + 2*t*K[3] + 2*t*K[4] + 3*t^2*K[4]}, 
 
   {K[3] + 3*t*K[4]}, {K[4]}}]
;[o]





                                               2
                          2                 5 t  K[4]
K[1] + t K[2] + t K[3] + t  K[3] + t K[4] + --------- + 
                                                2
 
   3
  t  K[4]

                                2
K[2] + 2 t K[3] + 2 t K[4] + 3 t  K[4]

K[3] + 3 t K[4]

K[4]
:[font = input; initialization; preserveAspect; startGroup]
*)
D[y,t]-A.y//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0}, {0}, {0}, {0}}]
;[o]
0

0

0

0
:[font = input; initialization; preserveAspect]
*)
Problem 2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A={{0,1},   \
   {-13,6}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1}, {-13, 6}}]
;[o]
0     1

-13   6
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixExp[A t];
u=Transpose[{Array[Sum[%[[#,j]]K[j],{j,2}]&,2]}]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(1/2 - (3*I)/4)*E^((3 - 2*I)*t)*K[1] + 
 
     (1/2 + (3*I)/4)*E^((3 + 2*I)*t)*K[1] + 
 
     I/4*E^((3 - 2*I)*t)*K[2] - I/4*E^((3 + 2*I)*t)*K[2]}, 
 
   {(-13*I)/4*E^((3 - 2*I)*t)*K[1] + 
 
     (13*I)/4*E^((3 + 2*I)*t)*K[1] + 
 
     (1/2 + (3*I)/4)*E^((3 - 2*I)*t)*K[2] + 
 
     (1/2 - (3*I)/4)*E^((3 + 2*I)*t)*K[2]}}]
;[o]
 1   3 I   (3 - 2 I) t         1   3 I   (3 + 2 I) t
(- - ---) E            K[1] + (- + ---) E            K[1] + 
 2    4                        2    4
 
  I  (3 - 2 I) t        I  (3 + 2 I) t
  - E            K[2] - - E            K[2]
  4                     4

-13 I  (3 - 2 I) t        13 I  (3 + 2 I) t
----- E            K[1] + ---- E            K[1] + 
  4                        4
 
   1   3 I   (3 - 2 I) t         1   3 I   (3 + 2 I) t
  (- + ---) E            K[2] + (- - ---) E            K[2]
   2    4                        2    4
:[font = input; initialization; preserveAspect; startGroup]
*)
y=K[4]+Integrate[K[3]+Integrate[u[[1,1]]-4/13,t],t]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
(-2*t^2)/13 + (23/338 + (9*I)/676)*E^((3 - 2*I)*t)*K[1] + 
 
  (23/338 - (9*I)/676)*E^((3 + 2*I)*t)*K[1] + 
 
  (-3/169 + (5*I)/676)*E^((3 - 2*I)*t)*K[2] + 
 
  (-3/169 - (5*I)/676)*E^((3 + 2*I)*t)*K[2] + t*K[3] + K[4]
;[o]
    2
-2 t     23    9 I   (3 - 2 I) t
----- + (--- + ---) E            K[1] + 
 13      338   676
 
   23    9 I   (3 + 2 I) t
  (--- - ---) E            K[1] + 
   338   676
 
      3     5 I   (3 - 2 I) t
  (-(---) + ---) E            K[2] + 
     169    676
 
      3     5 I   (3 + 2 I) t
  (-(---) - ---) E            K[2] + t K[3] + K[4]
     169    676
:[font = input; initialization; preserveAspect; startGroup]
*)
D[y,{t,4}]-6D[y,{t,3}]+13D[y,{t,2}]+4//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect]
*)
Problem 3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M={{1/75,-1/150},     \
   {1/75,-1/300}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1/75, -1/150}, {1/75, -1/300}}]
;[o]
1           1
--       -(---)
75         150

1           1
--       -(---)
75         300
:[font = input; initialization; preserveAspect; startGroup]
*)
A=MatrixExp[M t];
Clear[K]
y=Transpose[{Array[Sum[A[[#,j]]K[j],{j,2}]&,2]}]//Expand;
r=y[[1]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{(E^(((3 - I*7^(1/2))*t)/600)*K[1])/2 + 
 
   ((5*I)/2*E^(((3 - I*7^(1/2))*t)/600)*K[1])/7^(1/2) + 
 
   (E^(((3 + I*7^(1/2))*t)/600)*K[1])/2 - 
 
   ((5*I)/2*E^(((3 + I*7^(1/2))*t)/600)*K[1])/7^(1/2) - 
 
   (2*I*E^(((3 - I*7^(1/2))*t)/600)*K[2])/7^(1/2) + 
 
   (2*I*E^(((3 + I*7^(1/2))*t)/600)*K[2])/7^(1/2)}
;[o]
  ((3 - I Sqrt[7]) t)/600
 E                        K[1]
{----------------------------- + 
               2
 
   5 I  ((3 - I Sqrt[7]) t)/600
   --- E                        K[1]
    2
   --------------------------------- + 
                Sqrt[7]
 
    ((3 + I Sqrt[7]) t)/600
   E                        K[1]
   ----------------------------- - 
                 2
 
   5 I  ((3 + I Sqrt[7]) t)/600
   --- E                        K[1]
    2
   --------------------------------- - 
                Sqrt[7]
 
        ((3 - I Sqrt[7]) t)/600
   2 I E                        K[2]
   --------------------------------- + 
                Sqrt[7]
 
        ((3 + I Sqrt[7]) t)/600
   2 I E                        K[2]
   ---------------------------------}
                Sqrt[7]
:[font = input; initialization; preserveAspect; startGroup]
*)
f=y[[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{(4*I*E^(((3 - I*7^(1/2))*t)/600)*K[1])/7^(1/2) - 
 
   (4*I*E^(((3 + I*7^(1/2))*t)/600)*K[1])/7^(1/2) + 
 
   (E^(((3 - I*7^(1/2))*t)/600)*K[2])/2 - 
 
   ((5*I)/2*E^(((3 - I*7^(1/2))*t)/600)*K[2])/7^(1/2) + 
 
   (E^(((3 + I*7^(1/2))*t)/600)*K[2])/2 + 
 
   ((5*I)/2*E^(((3 + I*7^(1/2))*t)/600)*K[2])/7^(1/2)}
;[o]
      ((3 - I Sqrt[7]) t)/600
 4 I E                        K[1]
{--------------------------------- - 
              Sqrt[7]
 
        ((3 + I Sqrt[7]) t)/600
   4 I E                        K[1]
   --------------------------------- + 
                Sqrt[7]
 
    ((3 - I Sqrt[7]) t)/600
   E                        K[2]
   ----------------------------- - 
                 2
 
   5 I  ((3 - I Sqrt[7]) t)/600
   --- E                        K[2]
    2
   --------------------------------- + 
                Sqrt[7]
 
    ((3 + I Sqrt[7]) t)/600
   E                        K[2]
   ----------------------------- + 
                 2
 
   5 I  ((3 + I Sqrt[7]) t)/600
   --- E                        K[2]
    2
   ---------------------------------}
                Sqrt[7]
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
LinearSolve[A/.{t->0},{1000,1000}];
K[1]=%[[1]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
1000
;[o]
1000
:[font = input; initialization; preserveAspect; startGroup]
*)
K[2]=%%[[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
1000
;[o]
1000
:[font = input; initialization; preserveAspect; startGroup]
*)
Re[r/.{t->{10,20,100}}]//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1067.764700186698003, 1137.667110988617783, 
 
    1756.977404441146643}}]
;[o]
1067.76   1137.67   1756.98
:[font = input; initialization; preserveAspect; startGroup]
*)
Re[f/.{t->{10,20,100}}]//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1102.795714846362742, 1211.249700731157621, 
 
    2288.913256528842839}}]
;[o]
1102.8    1211.25   2288.91
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
LinearSolve[A/.{t->0},{2000,1000}];
K[1]=%[[1]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
2000
;[o]
2000
:[font = input; initialization; preserveAspect; startGroup]
*)
K[2]=%%[[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
1000
;[o]
1000
:[font = input; initialization; preserveAspect; startGroup]
*)
Re[r/.{t->{10,20,100}}]//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2205.591429692725483, 2422.499401462315243, 
 
    4577.826513057685678}}]
;[o]
2205.59   2422.5    4577.83
:[font = input; initialization; preserveAspect; startGroup]
*)
Re[f/.{t->{10,20,100}}]//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1242.919773485021695, 1505.580059701316973, 
 
    4416.656664879627625}}]
;[o]
1242.92   1505.58   4416.66
:[font = input; initialization; preserveAspect]
*)
Problem 4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
k1=12; k2=10; k3=12;
M={{0,1,0,0},      \
   {-k1-k2,0,k2,0},\
   {0,0,0,1},      \
   {k2,0,-k2-k3,0}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1, 0, 0}, {-22, 0, 10, 0}, {0, 0, 0, 1}, 
 
   {10, 0, -22, 0}}]
;[o]
0     1     0     0

-22   0     10    0

0     0     0     1

10    0     -22   0
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[K]
A=MatrixExp[M t];
u=Transpose[{Array[Sum[%[[#,j]]K[j],{j,4}]&,4]}]//Expand;
b={10,0,12,0}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{10, 0, 12, 0}
;[o]
{10, 0, 12, 0}
:[font = input; initialization; preserveAspect; startGroup]
*)
LinearSolve[A/.{t->0},b];
Do[K[i]=%[[i]],{i,4}]
y={u[[1]],u[[3]]}//ComplexExpand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-Cos[4*2^(1/2)*t] + 11*Cos[2*3^(1/2)*t]}, 
 
   {Cos[4*2^(1/2)*t] + 11*Cos[2*3^(1/2)*t]}}]
;[o]
-Cos[4 Sqrt[2] t] + 11 Cos[2 Sqrt[3] t]

Cos[4 Sqrt[2] t] + 11 Cos[2 Sqrt[3] t]
:[font = input; initialization; preserveAspect]
*)
Problems on Linear Recurrence Systems
(*
:[font = input; initialization; preserveAspect]
*)
<<DiscreteMath`Master`
(*
:[font = input; initialization; preserveAspect]
*)
Problem 1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Unprotect[M];
M=1/600{{6,-3},   \
        {3,-2}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1/100, -1/200}, {1/200, -1/300}}]
;[o]
 1          1
---      -(---)
100        200

 1          1
---      -(---)
200        300
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Eigensystem[M][[2]]]//Together
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(4 - 7^(1/2))/3, (4 + 7^(1/2))/3}, {1, 1}}]
;[o]
4 - Sqrt[7]   4 + Sqrt[7]
-----------   -----------
     3             3



1             1
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].M.P//Together
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(2 - 7^(1/2))/600, 0}, {0, (2 + 7^(1/2))/600}}]
;[o]
2 - Sqrt[7]
-----------
    600       0

              2 + Sqrt[7]
              -----------
0                 600
:[font = input; initialization; preserveAspect; startGroup]
*)
r=Table[R[i][n]/.RSolve[R[i][n+1]==(T[[i,i]]//N)R[i][n],R[i][n],n],{i,2}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{(-0.001076252185107650984)^n*R[1][0]}, 
 
   {0.007742918851774317651^n*R[2][0]}}]
;[o]
             n
(-0.00107625)  R[1][0]

          n
0.00774292  R[2][0]
:[font = input; initialization; preserveAspect; startGroup]
*)
p=P.r
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{((-0.001076252185107650984)^n*(4 - 7^(1/2))*
 
        R[1][0])/3 + (0.007742918851774317651^n*
 
        (4 + 7^(1/2))*R[2][0])/3}, 
 
   {(-0.001076252185107650984)^n*R[1][0] + 
 
     0.007742918851774317651^n*R[2][0]}}]
;[o]
             n
(-0.00107625)  (4 - Sqrt[7]) R[1][0]
------------------------------------ + 
                 3
 
            n
  0.00774292  (4 + Sqrt[7]) R[2][0]
  ---------------------------------
                  3

             n                     n
(-0.00107625)  R[1][0] + 0.00774292  R[2][0]
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Solve[(p[[1]]/.{n->0})-1000000.0==0,R[1][0]]//Expand;
Solve[(p[[2]]/.{n->0}/.%)-1000000.0==0,R[2][0]]//Expand;
p[[1]]/.{n->{1,2,3,4,5,6,7,8,9,10}}/.%%/.%//N//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{{5000., 41.66666666666666667, 0.3194444444444444445, 
 
     0.002476851851851851852, 0.00001917438271604938272, 
 
     1.484696502057613169*10^-7, 1.149584190672153635*10^-9, 
 
     8.90114168952903521*10^-12, 
 
     6.892081285246151503*10^-14, 
 
     5.336482664291520603*10^-16}}}}
;[o]
{{{{5000., 41.6667, 0.319444, 0.00247685, 0.0000191744, 
 
              -7            -9            -12
     1.4847 10  , 1.14958 10  , 8.90114 10   , 
 
               -14            -16
     6.89208 10   , 5.33648 10   }}}}
:[font = input; initialization; preserveAspect; startGroup]
*)
p[[2]]/.{n->{1,2,3,4,5,6,7,8,9,10}}/.%%%/.%%//N//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{{1666.666666666666667, 19.44444444444444444, 
 
     0.1435185185185185185, 0.001118827160493827161, 
 
     8.654835390946502059*10^-6, 6.702246227709190673*10^-8, 
 
     5.189400434385002287*10^-10, 
 
     4.018120808565767414*10^-12, 3.1111972419092618*10^-14, 
 
     2.408974895319988485*10^-16}}}}
;[o]
                                                      -6
{{{{1666.67, 19.4444, 0.143519, 0.00111883, 8.65484 10  , 
 
               -8           -10            -12
     6.70225 10  , 5.1894 10   , 4.01812 10   , 
 
              -14            -16
     3.1112 10   , 2.40897 10   }}}}
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect]
*)
Looking at the expressions for p[[1]] and p[[2]] as functions
of n above, we see that both are exponentials of basis < 1 in n
and therefore the populations will go extinct quite rapidly,
no matter the initial numbers.
(*
:[font = input; initialization; preserveAspect]
*)
Problem 2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M={{1,1,0},   \
   {-1,2,1},  \
   {-1,-1,4}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1, 0}, {-1, 2, 1}, {-1, -1, 4}}]
;[o]
1    1    0

-1   2    1

-1   -1   4
:[font = input; initialization; preserveAspect; startGroup]
*)
EV=Eigensystem[M][[2]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1, 1}, {0, 0, 0}, {1, 2, 3}}]
;[o]
1   1   1

0   0   0

1   2   3
:[font = input; initialization; preserveAspect; startGroup]
*)
NullSpace[EV]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, -2, 1}}]
;[o]
1    -2   1
:[font = input; initialization; preserveAspect; startGroup]
*)
P=Transpose[Join[{EV[[1]]},{EV[[3]]},%]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1, 1}, {1, 2, -2}, {1, 3, 1}}]
;[o]
1    1    1

1    2    -2

1    3    1
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Inverse[P].M.P//Together
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 0, -6}, {0, 3, 3}, {0, 0, 2}}]
;[o]
2    0    -6

0    3    3

0    0    2
:[font = input; initialization; preserveAspect]
*)
Having now upper triangularized our system of recurrence 
equations, we "RSolve" it in a do-loop in the order 3,2,1.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
r=Array[0&,3];
Do[r[[4-i]]=R[4-i][n]/.RSolve[R[4-i][n+1]==T[[4-i,4-i]]R[4-i][n] \
                             +Sum[T[[4-i,j]]r[[j,1]],{j,5-i,3}], \
                                             R[4-i][n],n],{i,3}]
r//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2^n*R[1][0] - 3*2^n*n*R[3][0]}, 
 
   {3^n*R[2][0] - 3*2^n*R[3][0] + 3*3^n*R[3][0]}, 
 
   {2^n*R[3][0]}}]
;[o]
 n              n
2  R[1][0] - 3 2  n R[3][0]

 n              n              n
3  R[2][0] - 3 2  R[3][0] + 3 3  R[3][0]

 n
2  R[3][0]
:[font = input; initialization; preserveAspect; startGroup]
*)
p=P.r//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2^n*R[1][0] + 3^n*R[2][0] - 2*2^n*R[3][0] + 
 
     3*3^n*R[3][0] - 3*2^n*n*R[3][0]}, 
 
   {2^n*R[1][0] + 2*3^n*R[2][0] - 8*2^n*R[3][0] + 
 
     6*3^n*R[3][0] - 3*2^n*n*R[3][0]}, 
 
   {2^n*R[1][0] + 3*3^n*R[2][0] - 8*2^n*R[3][0] + 
 
     9*3^n*R[3][0] - 3*2^n*n*R[3][0]}}]
;[o]
 n            n              n              n
2  R[1][0] + 3  R[2][0] - 2 2  R[3][0] + 3 3  R[3][0] - 
 
     n
  3 2  n R[3][0]

 n              n              n              n
2  R[1][0] + 2 3  R[2][0] - 8 2  R[3][0] + 6 3  R[3][0] - 
 
      n
   3 2  n R[3][0]

 n              n              n              n
2  R[1][0] + 3 3  R[2][0] - 8 2  R[3][0] + 9 3  R[3][0] - 
 
      n
   3 2  n R[3][0]
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[R]; R[1][0]:=1; R[2][0]:=1; R[3][0]:=1;
p/.{n->0}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{3}, {1}, {5}}]
;[o]
3

1

5
:[font = input; initialization; preserveAspect; startGroup]
*)
p/.{n->10}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{204452}, {434504}, {670700}}]
;[o]
204452

434504

670700
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[R]; R[1][0]:=1; R[2][0]:=1; R[3][0]:=-1;
p/.{n->0}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1}, {5}, {3}}]
;[o]
1

5

3
:[font = input; initialization; preserveAspect; startGroup]
*)
p/.{n->10}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-84306}, {-196260}, {-314358}}]
;[o]
-84306

-196260

-314358
:[font = input; initialization; preserveAspect]
*)
As one can guess from the above tests, the future is dependent 
on the initial populations. For instance, for p[0]={3,1,5}, the
populations would subsequenly grow indefinitely, whereis for
p[0]={1,5,3} they go extinct.
(*
:[font = input; initialization; preserveAspect]
*)
Exercises on Quadratics
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[x]; 
X=Array[Subscripted[x[#]]&,2]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{Subscripted[x[1]], Subscripted[x[2]]}
;[o]
{x , x }
  1   2
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[y]; 
Y=Array[Subscripted[y[#]]&,2]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{Subscripted[y[1]], Subscripted[y[2]]}
;[o]
{y , y }
  1   2
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M={{3,-4},{-4,-3}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{3, -4}, {-4, -3}}]
;[o]
3    -4

-4   -3
:[font = input; initialization; preserveAspect; startGroup]
*)
X.M.X//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
3*Subscripted[x[1]]^2 - 8*Subscripted[x[1]]*
 
   Subscripted[x[2]] - 3*Subscripted[x[2]]^2
;[o]
    2                 2
3 x   - 8 x  x  - 3 x
   1       1  2      2
:[font = input; initialization; preserveAspect; startGroup]
*)
S=Solve[%-1==0,X[[2]]]//Simplify
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{Subscripted[x[2]] -> 
 
     (-8*Subscripted[x[1]] - 
 
        (-12 + 100*Subscripted[x[1]]^2)^(1/2))/6}, 
 
   {Subscripted[x[2]] -> 
 
     (-8*Subscripted[x[1]] + 
 
        (-12 + 100*Subscripted[x[1]]^2)^(1/2))/6}}]
;[o]
                               2
      -8 x  - Sqrt[-12 + 100 x  ]
          1                   1
x  -> ---------------------------
 2                 6

                               2
      -8 x  + Sqrt[-12 + 100 x  ]
          1                   1
x  -> ---------------------------
 2                 6
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[f]
f[i_][x1_]:=(X[[2]]/.S[[i]])/.{X[[1]]->x1}
P1=Plot[{f[1][x1],f[2][x1]},{x1,-2,-Sqrt[12.001]/10}]
P2=Plot[{f[1][x1],f[2][x1]},{x1,Sqrt[12.001]/10,2}]
Show[P1,P2]
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;[o]
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;[o]
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c)
(*
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MatrixForm[{{5, 3}, {3, 3}}]
;[o]
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3   3
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5*Subscripted[x[1]]^2 + 6*Subscripted[x[1]]*
 
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;[o]
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5 x   + 6 x  x  + 3 x
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;[o]
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x  -> -x  - Sqrt[-] Sqrt[4 - x  ]
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;[o]
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(*
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;[o]
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(*
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                 2                    2
(4 - Sqrt[10]) y   + (4 + Sqrt[10]) y  =8
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d)
(*
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*)
M={{-1,0},    \
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(*
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MatrixForm[{{-1, 0}, {0, 1}}]
;[o]
-1   0

0    1
:[font = input; initialization; preserveAspect; startGroup]
*)
k={-2,0};
X.M.X+X.k
(*
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-2*Subscripted[x[1]] - Subscripted[x[1]]^2 + 
 
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;[o]
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Solve[%-1==0,X[[2]]]//Simplify
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;[o]
x  -> -1 - x
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Plot[{1+x1,-1-x1},{x1,-3,1}]
(*
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Graphics["<<>>"]
;[o]
-Graphics-
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*)
In order to write this one in the standard form, we need
the translation Y=X+{1,0}:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
X=Y-{1,0}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-1 + Subscripted[y[1]], Subscripted[y[2]]}
;[o]
{-1 + y , y }
       1   2
:[font = input; initialization; preserveAspect; startGroup]
*)
Print[(X.M.X+X.k-1)//Simplify,"=",1-1]
(*
:[font = print; inactive; preserveAspect; endGroup]
   2     2
-y   + y  =0
  1     2
:[font = input; initialization; preserveAspect]
*)
Exercise 2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
X=Array[Subscripted[x[#]]&,3]
Y=Array[Subscripted[y[#]]&,3]
(*
:[font = output; output; inactive; preserveAspect]
{Subscripted[x[1]], Subscripted[x[2]], Subscripted[x[3]]}
;[o]
{x , x , x }
  1   2   3
:[font = output; output; inactive; preserveAspect; endGroup]
{Subscripted[y[1]], Subscripted[y[2]], Subscripted[y[3]]}
;[o]
{y , y , y }
  1   2   3
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Table[If[i==j,1,-1],{i,3},{j,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, -1, -1}, {-1, 1, -1}, {-1, -1, 1}}]
;[o]
1    -1   -1

-1   1    -1

-1   -1   1
:[font = input; initialization; preserveAspect; startGroup]
*)
X.M.X//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
Subscripted[x[1]]^2 - 2*Subscripted[x[1]]*
 
   Subscripted[x[2]] + Subscripted[x[2]]^2 - 
 
  2*Subscripted[x[1]]*Subscripted[x[3]] - 
 
  2*Subscripted[x[2]]*Subscripted[x[3]] + Subscripted[x[3]]^2
;[o]
  2               2                         2
x   - 2 x  x  + x   - 2 x  x  - 2 x  x  + x
 1       1  2    2       1  3      2  3    3
:[font = input; initialization; preserveAspect; startGroup]
*)
S=Solve[%-3==0,X[[3]]]//Simplify
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{Subscripted[x[3]] -> 
 
     Subscripted[x[1]] + Subscripted[x[2]] - 
 
      (3 + 4*Subscripted[x[1]]*Subscripted[x[2]])^(1/2)}, 
 
   {Subscripted[x[3]] -> 
 
     Subscripted[x[1]] + Subscripted[x[2]] + 
 
      (3 + 4*Subscripted[x[1]]*Subscripted[x[2]])^(1/2)}}]
;[o]
x  -> x  + x  - Sqrt[3 + 4 x  x ]
 3     1    2               1  2

x  -> x  + x  + Sqrt[3 + 4 x  x ]
 3     1    2               1  2
:[font = input; initialization; preserveAspect; startGroup]
*)
f[i_][x1_,x2_]:=(X[[3]]/.S[[i]])/.{X[[1]]->x1}/.{X[[2]]->x2}
P1=Plot3D[f[1][x1,x2],{x1,-1/2,1/2},{x2,-3/2,3/2}]
P2=Plot3D[f[2][x1,x2],{x1,-1/2,1/2},{x2,-3/2,3/2}]
Show[P1,P2]
(*
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