(*^
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:[font = input; initialization; preserveAspect]
*)
Exercises on Differential Operators
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1
(*
:[font = input; initialization; preserveAspect]
*)
L[i_]:=If[i==1,D[f[x],x]-f[x],D[L[i-1],x]-i L[i-1]]
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[DSolve[{L[5]==0,{f[0],f'[0],f''[0],f'''[0],f''''[0]} \
                     ==IdentityMatrix[5][[i]]},f[x],x]],{i,5}]
(*
:[font = message; inactive; preserveAspect]
DSolve::dsdeg: 
   Warning: Differential equation of order higher than four
     encountered. DSolve may not be able to find the
     solution.
:[font = print; inactive; preserveAspect; endGroup]
             x       2 x       3 x      4 x    5 x
{{f[x] -> 5 E  - 10 E    + 10 E    - 5 E    + E   }}
               x        2 x       3 x       4 x       5 x
          -77 E    107 E      39 E      61 E      25 E
{{f[x] -> ------ + -------- - ------- + ------- - -------}}
            12        6          2         6        12
              x       2 x       3 x       4 x       5 x
          71 E    59 E      49 E      41 E      35 E
{{f[x] -> ----- - ------- + ------- - ------- + -------}}
           24        6         4         6        24
              x       2 x                4 x      5 x
          -7 E    13 E         3 x   11 E      5 E
{{f[x] -> ----- + ------- - 3 E    + ------- - ------}}
           12        6                  6        12
           x    2 x    3 x    4 x    5 x
          E    E      E      E      E
{{f[x] -> -- - ---- + ---- - ---- + ----}}
          24    6      4      6      24
:[font = input; initialization; preserveAspect]
*)
Exercise 2
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
DSolve[f''[x]-3f'[x]-4f[x]==e^x,f[x],x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{f[x] -> C[1]/E^x + E^(4*x)*C[2] + 
 
     e^x*(1/(5*(-4 + Log[e])) - 1/(5*(1 + Log[e])))}}
;[o]
          C[1]    4 x
{{f[x] -> ---- + E    C[2] + 
            x
           E
 
      x         1                1
     e  (--------------- - --------------)}}
         5 (-4 + Log[e])   5 (1 + Log[e])
:[font = input; initialization; preserveAspect; startGroup]
*)
DSolve[f''[x]-3f'[x]-4f[x]==0,f[x],x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{f[x] -> C[1]/E^x + E^(4*x)*C[2]}}
;[o]
          C[1]    4 x
{{f[x] -> ---- + E    C[2]}}
            x
           E
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
DSolve[f'[x]-4f[x]==0,f[x],x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{f[x] -> E^(4*x)*C[1]}}
;[o]
           4 x
{{f[x] -> E    C[1]}}
:[font = input; initialization; preserveAspect; startGroup]
*)
DSolve[f'[x]+f[x]==0,f[x],x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{f[x] -> C[1]/E^x}}
;[o]
          C[1]
{{f[x] -> ----}}
            x
           E
:[font = input; initialization; preserveAspect]
*)
We notice that the solution of the homogeneous equation
in part (a) is a linear combination of the solutions in
part (b).
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 3
(*
:[font = input; initialization; preserveAspect]
*)
Clear[f,L]
F[f_]:=D[f,x]-2f
L[f_]:=F[F[F[f]]]
(*
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L[E^(2x)]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect; startGroup]
*)
L[x E^(2x)]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect; startGroup]
*)
L[x^2 E^(2x)]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L[f[x]]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-8*f[x] + 12*Derivative[1][f][x] - 6*Derivative[2][f][x] + 
 
  Derivative[3][f][x]
;[o]
                                 (3)
-8 f[x] + 12 f'[x] - 6 f''[x] + f   [x]
:[font = input; initialization; preserveAspect; startGroup]
*)
f[x]/.DSolve[L[f[x]]==0,f[x],x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{E^(2*x)*C[1] + E^(2*x)*x*C[2] + E^(2*x)*x^2*C[3]}
;[o]
  2 x         2 x           2 x  2
{E    C[1] + E    x C[2] + E    x  C[3]}
:[font = input; initialization; preserveAspect; startGroup]
*)
B=Table[(%[[1]]-(%[[1]]/.{C[i]->0}))/.{C[i]->1},{i,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{E^(2*x), E^(2*x)*x, E^(2*x)*x^2}
;[o]
  2 x   2 x     2 x  2
{E   , E    x, E    x }
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Table[D[B[[j]],{x,i-1}],{i,3},{j,3}]; M//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{E^(2*x), E^(2*x)*x, E^(2*x)*x^2}, 
 
   {2*E^(2*x), E^(2*x) + 2*E^(2*x)*x, 
 
    2*E^(2*x)*x + 2*E^(2*x)*x^2}, 
 
   {4*E^(2*x), 4*E^(2*x) + 4*E^(2*x)*x, 
 
    2*E^(2*x) + 8*E^(2*x)*x + 4*E^(2*x)*x^2}}]
;[o]
 2 x      2 x                 2 x  2
E        E    x              E    x

   2 x    2 x      2 x          2 x        2 x  2
2 E      E    + 2 E    x     2 E    x + 2 E    x

   2 x      2 x      2 x        2 x      2 x        2 x  2
4 E      4 E    + 4 E    x   2 E    + 8 E    x + 4 E    x
:[font = input; initialization; preserveAspect; startGroup]
*)
W[0]=Det[M]/.{x->0}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
2
;[o]
2
:[font = input; initialization; preserveAspect]
*)
We can now conclude that the solutions in part (a) are
a basis for the kernel of L, taking also into account the
fact that the kernel has dimension 3. 
(*
:[font = input; initialization; preserveAspect]
*)
c)
We can conjecture that (x^i)E^(r x) , with 0 <= i < f ,
will be solutions.
(*
:[font = input; initialization; preserveAspect]
*)
L[f_]:=F[F[F[F[F[f]]]]]
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[L[x^i E^(2x)]//Expand],{i,0,4}]
(*
:[font = print; inactive; preserveAspect; endGroup]
0
0
0
0
0
:[font = input; initialization; preserveAspect]
*)
Exercise 4
a)
(*
:[font = input; initialization; preserveAspect]
*)
Clear[f,L]
F1[f_]:=D[f,x]+2f
F2[f_]:=F1[F1[f]]+3^2f
L[f_]:=F2[F2[F2[f]]]
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L[x^i E^(-2x)Cos[3x]]//Expand/.{i->{0,1,2}}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
(-120*i*x^(-6 + i)*Cos[3*x])/E^(2*x) + 
 
  (274*i^2*x^(-6 + i)*Cos[3*x])/E^(2*x) - 
 
  (225*i^3*x^(-6 + i)*Cos[3*x])/E^(2*x) + 
 
  (85*i^4*x^(-6 + i)*Cos[3*x])/E^(2*x) - 
 
  (15*i^5*x^(-6 + i)*Cos[3*x])/E^(2*x) + 
 
  (i^6*x^(-6 + i)*Cos[3*x])/E^(2*x) + 
 
  (648*i*x^(-4 + i)*Cos[3*x])/E^(2*x) - 
 
  (1188*i^2*x^(-4 + i)*Cos[3*x])/E^(2*x) + 
 
  (648*i^3*x^(-4 + i)*Cos[3*x])/E^(2*x) - 
 
  (108*i^4*x^(-4 + i)*Cos[3*x])/E^(2*x) - 
 
  (432*i*x^(-5 + i)*Sin[3*x])/E^(2*x) + 
 
  (900*i^2*x^(-5 + i)*Sin[3*x])/E^(2*x) - 
 
  (630*i^3*x^(-5 + i)*Sin[3*x])/E^(2*x) + 
 
  (180*i^4*x^(-5 + i)*Sin[3*x])/E^(2*x) - 
 
  (18*i^5*x^(-5 + i)*Sin[3*x])/E^(2*x) + 
 
  (432*i*x^(-3 + i)*Sin[3*x])/E^(2*x) - 
 
  (648*i^2*x^(-3 + i)*Sin[3*x])/E^(2*x) + 
 
  (216*i^3*x^(-3 + i)*Sin[3*x])/E^(2*x)
;[o]
        -6 + i                 2  -6 + i
-120 i x       Cos[3 x]   274 i  x       Cos[3 x]
----------------------- + ----------------------- - 
          2 x                       2 x
         E                         E
 
       3  -6 + i                4  -6 + i
  225 i  x       Cos[3 x]   85 i  x       Cos[3 x]
  ----------------------- + ---------------------- - 
            2 x                       2 x
           E                         E
 
      5  -6 + i             6  -6 + i
  15 i  x       Cos[3 x]   i  x       Cos[3 x]
  ---------------------- + ------------------- + 
            2 x                    2 x
           E                      E
 
         -4 + i                  2  -4 + i
  648 i x       Cos[3 x]   1188 i  x       Cos[3 x]
  ---------------------- - ------------------------ + 
            2 x                       2 x
           E                         E
 
       3  -4 + i                 4  -4 + i
  648 i  x       Cos[3 x]   108 i  x       Cos[3 x]
  ----------------------- - ----------------------- - 
            2 x                       2 x
           E                         E
 
         -5 + i                 2  -5 + i
  432 i x       Sin[3 x]   900 i  x       Sin[3 x]
  ---------------------- + ----------------------- - 
            2 x                      2 x
           E                        E
 
       3  -5 + i                 4  -5 + i
  630 i  x       Sin[3 x]   180 i  x       Sin[3 x]
  ----------------------- + ----------------------- - 
            2 x                       2 x
           E                         E
 
      5  -5 + i                   -3 + i
  18 i  x       Sin[3 x]   432 i x       Sin[3 x]
  ---------------------- + ---------------------- - 
            2 x                      2 x
           E                        E
 
       2  -3 + i                 3  -3 + i
  648 i  x       Sin[3 x]   216 i  x       Sin[3 x]
  ----------------------- + -----------------------
            2 x                       2 x
           E                         E
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L1[i_]:=L[x^i E^(-2x)Sin[3x]]//Expand
L2[i_]:=L[x^i E^(-2x)Cos[3x]]//Expand
Do[Print[L1[i],",",L2[i]],{i,0,2}]
(*
:[font = print; inactive; preserveAspect; endGroup]
0,0
0,0
0,0
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L[f[x]]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
2197*f[x] + 2028*Derivative[1][f][x] + 
 
  1131*Derivative[2][f][x] + 376*Derivative[3][f][x] + 
 
  87*Derivative[4][f][x] + 12*Derivative[5][f][x] + 
 
  Derivative[6][f][x]
;[o]
                                            (3)
2197 f[x] + 2028 f'[x] + 1131 f''[x] + 376 f   [x] + 
 
      (4)          (5)       (6)
  87 f   [x] + 12 f   [x] + f   [x]
:[font = input; initialization; preserveAspect; startGroup]
*)
f[x]/.DSolve[L[f[x]]==0,f[x],x]
(*
:[font = message; inactive; preserveAspect]
DSolve::dsdeg: 
   Warning: Differential equation of order higher than four
     encountered. DSolve may not be able to find the
     solution.
:[font = output; output; inactive; preserveAspect; endGroup]
{(C[2]*Cos[3*x])/E^(2*x) + (x*C[4]*Cos[3*x])/E^(2*x) + 
 
   (x^2*C[6]*Cos[3*x])/E^(2*x) - (C[1]*Sin[3*x])/E^(2*x) - 
 
   (x*C[3]*Sin[3*x])/E^(2*x) - (x^2*C[5]*Sin[3*x])/E^(2*x)}
;[o]
                                    2
 C[2] Cos[3 x]   x C[4] Cos[3 x]   x  C[6] Cos[3 x]
{------------- + --------------- + ---------------- - 
      2 x              2 x                2 x
     E                E                  E
 
                                      2
   C[1] Sin[3 x]   x C[3] Sin[3 x]   x  C[5] Sin[3 x]
   ------------- - --------------- - ----------------}
        2 x              2 x                2 x
       E                E                  E
:[font = input; initialization; preserveAspect; startGroup]
*)
B=Table[(%[[1]]-(%[[1]]/.{C[i]->0}))/.{C[i]->1},{i,6}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-(Sin[3*x]/E^(2*x)), Cos[3*x]/E^(2*x), 
 
  -((x*Sin[3*x])/E^(2*x)), (x*Cos[3*x])/E^(2*x), 
 
  -((x^2*Sin[3*x])/E^(2*x)), (x^2*Cos[3*x])/E^(2*x)}
;[o]
   Sin[3 x]   Cos[3 x]    x Sin[3 x]   x Cos[3 x]
{-(--------), --------, -(----------), ----------, 
      2 x        2 x          2 x          2 x
     E          E            E            E
 
     2             2
    x  Sin[3 x]   x  Cos[3 x]
  -(-----------), -----------}
        2 x           2 x
       E             E
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Table[D[B[[j]],{x,i-1}],{i,3},{j,3}]; M//TableForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
TableForm[{{-(Sin[3*x]/E^(2*x)), Cos[3*x]/E^(2*x), 
 
    -((x*Sin[3*x])/E^(2*x))}, 
 
   {(-3*Cos[3*x])/E^(2*x) + (2*Sin[3*x])/E^(2*x), 
 
    (-2*Cos[3*x])/E^(2*x) - (3*Sin[3*x])/E^(2*x), 
 
    (-3*x*Cos[3*x])/E^(2*x) - Sin[3*x]/E^(2*x) + 
 
     (2*x*Sin[3*x])/E^(2*x)}, 
 
   {(12*Cos[3*x])/E^(2*x) + (5*Sin[3*x])/E^(2*x), 
 
    (-5*Cos[3*x])/E^(2*x) + (12*Sin[3*x])/E^(2*x), 
 
    (-6*Cos[3*x])/E^(2*x) + (12*x*Cos[3*x])/E^(2*x) + 
 
     (4*Sin[3*x])/E^(2*x) + (5*x*Sin[3*x])/E^(2*x)}}]
;[o]
  Sin[3 x]                 Cos[3 x]
-(--------)                --------
     2 x                      2 x
    E                        E
 
     x Sin[3 x]
   -(----------)
         2 x
        E

-3 Cos[3 x]   2 Sin[3 x]   -2 Cos[3 x]   3 Sin[3 x]
----------- + ----------   ----------- - ----------
    2 x           2 x          2 x           2 x
   E             E            E             E
 
   -3 x Cos[3 x]   Sin[3 x]   2 x Sin[3 x]
   ------------- - -------- + ------------
        2 x           2 x          2 x
       E             E            E

12 Cos[3 x]   5 Sin[3 x]   -5 Cos[3 x]   12 Sin[3 x]
----------- + ----------   ----------- + -----------
    2 x           2 x          2 x           2 x
   E             E            E             E
 
   -6 Cos[3 x]   12 x Cos[3 x]   4 Sin[3 x]   5 x Sin[3 x]
   ----------- + ------------- + ---------- + ------------
       2 x            2 x            2 x           2 x
      E              E              E             E
:[font = input; initialization; preserveAspect; startGroup]
*)
W[0]=Det[M]/.{x->0}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-18
;[o]
-18
:[font = input; initialization; preserveAspect]
*)
d)
They are the same set of six linearly independent solutions.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 5
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[f,L]
L[f_]:=D[f,{x,3}]-6D[f,{x,2}]+11D[f,x]-6f
L[f[x]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-6*f[x] + 11*Derivative[1][f][x] - 6*Derivative[2][f][x] + 
 
  Derivative[3][f][x]
;[o]
                                 (3)
-6 f[x] + 11 f'[x] - 6 f''[x] + f   [x]
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
f[x]/.DSolve[L[f[x]]==0,f[x],x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{E^x*C[1] + E^(2*x)*C[2] + E^(3*x)*C[3]}
;[o]
  x         2 x         3 x
{E  C[1] + E    C[2] + E    C[3]}
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
B=Table[(%[[1]]-(%[[1]]/.{C[i]->0}))/.{C[i]->1},{i,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{E^x, E^(2*x), E^(3*x)}
;[o]
  x   2 x   3 x
{E , E   , E   }
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
f[x]/.DSolve[L[f[x]]==E^(3x),f[x],x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{(E^(3*x)*x)/2 + E^x*C[1] + E^(2*x)*C[2] + E^(3*x)*C[3]}
;[o]
  3 x
 E    x    x         2 x         3 x
{------ + E  C[1] + E    C[2] + E    C[3]}
   2
:[font = input; initialization; preserveAspect]
*)
Exercise 6
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
p=x^5+x^4+x^3+x^2+x+1
(*
:[font = output; output; inactive; preserveAspect; endGroup]
1 + x + x^2 + x^3 + x^4 + x^5
;[o]
         2    3    4    5
1 + x + x  + x  + x  + x
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[L,f]
L[f_]:=Sum[Coefficient[p,x^i]D[f,{x,i}],{i,0,5}]
L[f[x]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
f[x] + Derivative[1][f][x] + Derivative[2][f][x] + 
 
  Derivative[3][f][x] + Derivative[4][f][x] + 
 
  Derivative[5][f][x]
;[o]
                         (3)       (4)       (5)
f[x] + f'[x] + f''[x] + f   [x] + f   [x] + f   [x]
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
p//Factor
(*
:[font = output; output; inactive; preserveAspect; endGroup]
(1 + x)*(1 - x + x^2)*(1 + x + x^2)
;[o]
                  2            2
(1 + x) (1 - x + x ) (1 + x + x )
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
f[x]/.DSolve[L[f[x]]==0,f[x],x]
(*
:[font = message; inactive; preserveAspect]
DSolve::dsdeg: 
   Warning: Differential equation of order higher than four
     encountered. DSolve may not be able to find the
     solution.
:[font = output; output; inactive; preserveAspect; endGroup]
{C[1]/E^x + E^(((-1 - I*3^(1/2))*x)/2)*C[2] + 
 
   E^(((1 - I*3^(1/2))*x)/2)*C[3] + 
 
   E^(((-1 + I*3^(1/2))*x)/2)*C[4] + 
 
   E^(((1 + I*3^(1/2))*x)/2)*C[5]}
;[o]
 C[1]    ((-1 - I Sqrt[3]) x)/2
{---- + E                       C[2] + 
   x
  E
 
    ((1 - I Sqrt[3]) x)/2
   E                      C[3] + 
 
    ((-1 + I Sqrt[3]) x)/2         ((1 + I Sqrt[3]) x)/2
   E                       C[4] + E                      C[5]
 
   }
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
f[x]/.DSolve[{L[f[x]]==0,{f'[0],f''[0],f'''[0],f''''[0]}=={0,0,0,0}},f[x],x]
(*
:[font = message; inactive; preserveAspect]
DSolve::dsdeg: 
   Warning: Differential equation of order higher than four
     encountered. DSolve may not be able to find the
     solution.
:[font = message; inactive; preserveAspect]
Solve::svars: 
   Warning: Equations may not give solutions for all "solve"
     variables.
:[font = output; output; inactive; preserveAspect; endGroup]
{(4*I*C[5])/((I + 3^(1/2))*E^x) + 
 
   I*3^(1/2)*E^(((-1 - I*3^(1/2))*x)/2)*C[5] + 
 
   ((I - 3^(1/2))*E^(((1 - I*3^(1/2))*x)/2)*C[5])/
 
    (I + 3^(1/2)) + ((3*I + 3^(1/2))*
 
      E^(((-1 + I*3^(1/2))*x)/2)*C[5])/(I + 3^(1/2)) + 
 
   E^(((1 + I*3^(1/2))*x)/2)*C[5]}
;[o]
     4 I C[5]                  ((-1 - I Sqrt[3]) x)/2
{---------------- + I Sqrt[3] E                       C[5] + 
                x
 (I + Sqrt[3]) E
 
                  ((1 - I Sqrt[3]) x)/2
   (I - Sqrt[3]) E                      C[5]
   ----------------------------------------- + 
                  I + Sqrt[3]
 
                    ((-1 + I Sqrt[3]) x)/2
   (3 I + Sqrt[3]) E                       C[5]
   -------------------------------------------- + 
                   I + Sqrt[3]
 
    ((1 + I Sqrt[3]) x)/2
   E                      C[5]}
:[font = input; initialization; preserveAspect]
*)
d)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
DSolve[{L[f[x]]==0,{f[0],f'[0],f''[0],f'''[0],f''''[0]} \
                   =={0,0,0,0,0}},f[x],x]
(*
:[font = message; inactive; preserveAspect]
DSolve::dsdeg: 
   Warning: Differential equation of order higher than four
     encountered. DSolve may not be able to find the
     solution.
:[font = output; output; inactive; preserveAspect; endGroup]
{{f[x] -> 0}}
;[o]
{{f[x] -> 0}}
:[font = input; initialization; preserveAspect]
*)
Check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
f[x]=%%/.Solve[(%%/.{x->0})==0,C[5]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{0}}
;[o]
{{0}}
:[font = input; initialization; preserveAspect]
*)
Exercise 7
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[f,F,L]
F[f_]:=D[f,{x,3}]+D[f,{x,2}]+f
L[f_]:=F[F[f]]
L[f[x]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
f[x] + 2*Derivative[2][f][x] + 2*Derivative[3][f][x] + 
 
  Derivative[4][f][x] + 2*Derivative[5][f][x] + 
 
  Derivative[6][f][x]
;[o]
                     (3)       (4)         (5)       (6)
f[x] + 2 f''[x] + 2 f   [x] + f   [x] + 2 f   [x] + f   [x]
:[font = input; initialization; preserveAspect; startGroup]
*)
p=Sum[Coefficient[L[f[x]],D[f[x],{x,i}]]x^i,{i,0,6}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
1 + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + x^6
;[o]
       2      3    4      5    6
1 + 2 x  + 2 x  + x  + 2 x  + x
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
p//Factor
(*
:[font = output; output; inactive; preserveAspect; endGroup]
(1 + x^2 + x^3)^2
;[o]
      2    3 2
(1 + x  + x )
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
s=x/.Solve[p==0,x];
sR=Re[s]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-1/3 + 1/(3*2^(2/3)*(29 - 3*93^(1/2))^(1/3)) + 
 
   (29 - 3*93^(1/2))^(1/3)/(6*2^(1/3)), 
 
  -1/3 + 1/(3*2^(2/3)*(29 - 3*93^(1/2))^(1/3)) + 
 
   (29 - 3*93^(1/2))^(1/3)/(6*2^(1/3)), 
 
  -1/3 - 2^(1/3)/(3*(29 - 3*93^(1/2))^(1/3)) - 
 
   (29 - 3*93^(1/2))^(1/3)/(3*2^(1/3)), 
 
  -1/3 - 2^(1/3)/(3*(29 - 3*93^(1/2))^(1/3)) - 
 
   (29 - 3*93^(1/2))^(1/3)/(3*2^(1/3)), 
 
  -1/3 + 1/(3*2^(2/3)*(29 - 3*93^(1/2))^(1/3)) + 
 
   (29 - 3*93^(1/2))^(1/3)/(6*2^(1/3)), 
 
  -1/3 + 1/(3*2^(2/3)*(29 - 3*93^(1/2))^(1/3)) + 
 
   (29 - 3*93^(1/2))^(1/3)/(6*2^(1/3))}
;[o]
                                                       1/3
   1                 1                (29 - 3 Sqrt[93])
{-(-) + --------------------------- + --------------------, 
   3       2/3                  1/3             1/3
        3 2    (29 - 3 Sqrt[93])             6 2
 
                                                        1/3
    1                 1                (29 - 3 Sqrt[93])
  -(-) + --------------------------- + --------------------, 
    3       2/3                  1/3             1/3
         3 2    (29 - 3 Sqrt[93])             6 2
 
                   1/3                             1/3
    1             2               (29 - 3 Sqrt[93])
  -(-) - ---------------------- - --------------------, 
    3                       1/3             1/3
         3 (29 - 3 Sqrt[93])             3 2
 
                   1/3                             1/3
    1             2               (29 - 3 Sqrt[93])
  -(-) - ---------------------- - --------------------, 
    3                       1/3             1/3
         3 (29 - 3 Sqrt[93])             3 2
 
                                                        1/3
    1                 1                (29 - 3 Sqrt[93])
  -(-) + --------------------------- + --------------------, 
    3       2/3                  1/3             1/3
         3 2    (29 - 3 Sqrt[93])             6 2
 
                                                        1/3
    1                 1                (29 - 3 Sqrt[93])
  -(-) + --------------------------- + --------------------}
    3       2/3                  1/3             1/3
         3 2    (29 - 3 Sqrt[93])             6 2
:[font = input; initialization; preserveAspect; startGroup]
*)
sI=Im[s]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-(1/(2^(2/3)*3^(1/2)*(29 - 3*93^(1/2))^(1/3))) + 
 
   (29 - 3*93^(1/2))^(1/3)/(2*2^(1/3)*3^(1/2)), 
 
  -(1/(2^(2/3)*3^(1/2)*(29 - 3*93^(1/2))^(1/3))) + 
 
   (29 - 3*93^(1/2))^(1/3)/(2*2^(1/3)*3^(1/2)), 0, 0, 
 
  1/(2^(2/3)*3^(1/2)*(29 - 3*93^(1/2))^(1/3)) - 
 
   (29 - 3*93^(1/2))^(1/3)/(2*2^(1/3)*3^(1/2)), 
 
  1/(2^(2/3)*3^(1/2)*(29 - 3*93^(1/2))^(1/3)) - 
 
   (29 - 3*93^(1/2))^(1/3)/(2*2^(1/3)*3^(1/2))}
;[o]
                   1
{-(---------------------------------) + 
    2/3                          1/3
   2    Sqrt[3] (29 - 3 Sqrt[93])
 
                    1/3
   (29 - 3 Sqrt[93])
   --------------------, 
         1/3
      2 2    Sqrt[3]
 
                    1
  -(---------------------------------) + 
     2/3                          1/3
    2    Sqrt[3] (29 - 3 Sqrt[93])
 
                    1/3
   (29 - 3 Sqrt[93])
   --------------------, 0, 0, 
         1/3
      2 2    Sqrt[3]
 
                                                       1/3
                  1                   (29 - 3 Sqrt[93])
  --------------------------------- - --------------------, 
   2/3                          1/3         1/3
  2    Sqrt[3] (29 - 3 Sqrt[93])         2 2    Sqrt[3]
 
                                                       1/3
                  1                   (29 - 3 Sqrt[93])
  --------------------------------- - --------------------}
   2/3                          1/3         1/3
  2    Sqrt[3] (29 - 3 Sqrt[93])         2 2    Sqrt[3]
:[font = input; initialization; preserveAspect]
*)
We will have to use floating point representations here,
otherwise the result gets too complicated for Mathematica.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
j=Mod[2i,6]+1;
f=Sum[If[i<4,C[i]E^(sR[[j]]x)(Cos[sI[[j]]x]+I Sin[sI[[j]]x]),       \
             x C[i]E^(sR[[j]]x)(Cos[sI[[j]]x]+I Sin[sI[[j]]x])],{i,6}];
f//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
C[1]/2.718281828459045235^(1.46557123187676803*x) + 
 
  (x*C[4])/2.718281828459045235^(1.46557123187676803*x) + 
 
  2.718281828459045235^(0.2327856159383840151*x)*C[3]*
 
   (Cos[0.7925519925154478522*x] - 
 
     1.*I*Sin[0.7925519925154478522*x]) + 
 
  2.718281828459045235^(0.2327856159383840151*x)*x*C[6]*
 
   (Cos[0.7925519925154478522*x] - 
 
     1.*I*Sin[0.7925519925154478522*x]) + 
 
  2.718281828459045235^(0.2327856159383840151*x)*C[2]*
 
   (Cos[0.7925519925154478522*x] + 
 
     1.*I*Sin[0.7925519925154478522*x]) + 
 
  2.718281828459045235^(0.2327856159383840151*x)*x*C[5]*
 
   (Cos[0.7925519925154478522*x] + 
 
     1.*I*Sin[0.7925519925154478522*x])
;[o]
      C[1]              x C[4]
---------------- + ---------------- + 
       1.46557 x          1.46557 x
2.71828            2.71828
 
         0.232786 x
  2.71828           C[3] 
 
   (Cos[0.792552 x] - 1. I Sin[0.792552 x]) + 
 
         0.232786 x
  2.71828           x C[6] 
 
   (Cos[0.792552 x] - 1. I Sin[0.792552 x]) + 
 
         0.232786 x
  2.71828           C[2] 
 
   (Cos[0.792552 x] + 1. I Sin[0.792552 x]) + 
 
         0.232786 x
  2.71828           x C[5] 
 
   (Cos[0.792552 x] + 1. I Sin[0.792552 x])
:[font = input; initialization; preserveAspect]
*)
Solution check (gives a floating point approximation
close enough to 0):
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L[f//N]/.{x->5}//Simplify
(*
:[font = output; output; inactive; preserveAspect; endGroup]
1.146676760991513661*10^-37*C[1] + 
 
  (-(1.301042606982605321*10^-18) - 
 
     8.673617379884035472*10^-19*I)*C[2] + 
 
  (-(1.301042606982605321*10^-18) + 
 
     8.673617379884035472*10^-19*I)*C[3] - 
 
  6.182761645643729848*10^-20*C[4] + 
 
  (-(2.289834988289385365*10^-16) + 
 
     7.112366251504909087*10^-17*I)*C[5] + 
 
  (-(2.289834988289385365*10^-16) - 
 
     7.112366251504909087*10^-17*I)*C[6]
;[o]
          -37                    -18             -19
1.14668 10    C[1] + (-1.30104 10    - 8.67362 10    I) 
 
                      -18             -19
   C[2] + (-1.30104 10    + 8.67362 10    I) C[3] - 
 
            -20                    -16             -17
  6.18276 10    C[4] + (-2.28983 10    + 7.11237 10    I) 
 
                      -16             -17
   C[5] + (-2.28983 10    - 7.11237 10    I) C[6]
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
x/.Solve[p==1,x]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-1, 0, 0, -1/3 + 2^(1/3)/(3*(-56 + 6*87^(1/2))^(1/3)) + 
 
   (-56 + 6*87^(1/2))^(1/3)/(3*2^(1/3)), 
 
  -1/3 - (1 + I*3^(1/2))/
 
    (3*2^(2/3)*(-56 + 6*87^(1/2))^(1/3)) - 
 
   ((1 - I*3^(1/2))*(-56 + 6*87^(1/2))^(1/3))/(6*2^(1/3)), 
 
  -1/3 - (1 - I*3^(1/2))/
 
    (3*2^(2/3)*(-56 + 6*87^(1/2))^(1/3)) - 
 
   ((1 + I*3^(1/2))*(-56 + 6*87^(1/2))^(1/3))/(6*2^(1/3))}
;[o]
                            1/3
             1             2
{-1, 0, 0, -(-) + ----------------------- + 
             3                        1/3
                  3 (-56 + 6 Sqrt[87])
 
                     1/3
   (-56 + 6 Sqrt[87])
   ---------------------, 
             1/3
          3 2
 
    1           1 + I Sqrt[3]
  -(-) - ---------------------------- - 
    3       2/3                   1/3
         3 2    (-56 + 6 Sqrt[87])
 
                                     1/3
   (1 - I Sqrt[3]) (-56 + 6 Sqrt[87])
   -------------------------------------, 
                     1/3
                  6 2
 
    1           1 - I Sqrt[3]
  -(-) - ---------------------------- - 
    3       2/3                   1/3
         3 2    (-56 + 6 Sqrt[87])
 
                                     1/3
   (1 + I Sqrt[3]) (-56 + 6 Sqrt[87])
   -------------------------------------}
                     1/3
                  6 2
:[font = input; initialization; preserveAspect]
*)
We notice from above that f=E^(-x)is a particular 
solution for the nonhomogeneous equation.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
f=f+E^(-x);
f//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
2.718281828459045235^(-1.*x) + 
 
  C[1]/2.718281828459045235^(1.46557123187676803*x) + 
 
  (x*C[4])/2.718281828459045235^(1.46557123187676803*x) + 
 
  2.718281828459045235^(0.2327856159383840151*x)*C[3]*
 
   (Cos[0.7925519925154478522*x] - 
 
     1.*I*Sin[0.7925519925154478522*x]) + 
 
  2.718281828459045235^(0.2327856159383840151*x)*x*C[6]*
 
   (Cos[0.7925519925154478522*x] - 
 
     1.*I*Sin[0.7925519925154478522*x]) + 
 
  2.718281828459045235^(0.2327856159383840151*x)*C[2]*
 
   (Cos[0.7925519925154478522*x] + 
 
     1.*I*Sin[0.7925519925154478522*x]) + 
 
  2.718281828459045235^(0.2327856159383840151*x)*x*C[5]*
 
   (Cos[0.7925519925154478522*x] + 
 
     1.*I*Sin[0.7925519925154478522*x])
;[o]
       -1. x         C[1]              x C[4]
2.71828      + ---------------- + ---------------- + 
                      1.46557 x          1.46557 x
               2.71828            2.71828
 
         0.232786 x
  2.71828           C[3] 
 
   (Cos[0.792552 x] - 1. I Sin[0.792552 x]) + 
 
         0.232786 x
  2.71828           x C[6] 
 
   (Cos[0.792552 x] - 1. I Sin[0.792552 x]) + 
 
         0.232786 x
  2.71828           C[2] 
 
   (Cos[0.792552 x] + 1. I Sin[0.792552 x]) + 
 
         0.232786 x
  2.71828           x C[5] 
 
   (Cos[0.792552 x] + 1. I Sin[0.792552 x])
:[font = input; initialization; preserveAspect; startGroup]
*)
Solve[Table[D[f//N,{x,i-1}]/.{x->0},{i,6}]=={0,0,1,0,0,0}, \
                             {C[1],C[2],C[3],C[4],C[5],C[6]}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{C[4] -> -0.03773461807958688222 + 
 
     8.470329472543003391*10^-21*I, 
 
   C[5] -> -0.1585520457989162356 + 
 
     0.04937196153158245367*I, 
 
   C[6] -> -0.1585520457989162356 - 
 
     0.04937196153158245371*I, 
 
   C[1] -> -0.3187290896402679518 + 
 
     2.032879073410320814*10^-20*I, 
 
   C[2] -> -0.3406354551798660241 - 0.660088299258783936*I, 
 
   C[3] -> -0.3406354551798660241 + 0.6600882992587839361*I}}
;[o]
                                 -21
{{C[4] -> -0.0377346 + 8.47033 10    I, 
 
   C[5] -> -0.158552 + 0.049372 I, 
 
   C[6] -> -0.158552 - 0.049372 I, 
 
                                 -20
   C[1] -> -0.318729 + 2.03288 10    I, 
 
   C[2] -> -0.340635 - 0.660088 I, 
 
   C[3] -> -0.340635 + 0.660088 I}}
:[font = input; initialization; preserveAspect; startGroup]
*)
(f//N)/.%
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{(-0.3187290896402679518 + 2.032879073410320814*10^-20*I)/
 
    2.718281828459045235^(1.46557123187676803*x) + 
 
   2.718281828459045235^(-1.*x) + 
 
   ((-0.03773461807958688222 + 
 
        8.470329472543003391*10^-21*I)*x)/
 
    2.718281828459045235^(1.46557123187676803*x) + 
 
   (-0.3406354551798660241 + 0.6600882992587839361*I)*
 
    2.718281828459045235^(0.2327856159383840151*x)*
 
    (Cos[0.7925519925154478522*x] - 
 
      1.*I*Sin[0.7925519925154478522*x]) + 
 
   (-0.1585520457989162356 - 0.04937196153158245371*I)*
 
    2.718281828459045235^(0.2327856159383840151*x)*x*
 
    (Cos[0.7925519925154478522*x] - 
 
      1.*I*Sin[0.7925519925154478522*x]) + 
 
   (-0.3406354551798660241 - 0.660088299258783936*I)*
 
    2.718281828459045235^(0.2327856159383840151*x)*
 
    (Cos[0.7925519925154478522*x] + 
 
      1.*I*Sin[0.7925519925154478522*x]) + 
 
   (-0.1585520457989162356 + 0.04937196153158245367*I)*
 
    2.718281828459045235^(0.2327856159383840151*x)*x*
 
    (Cos[0.7925519925154478522*x] + 
 
      1.*I*Sin[0.7925519925154478522*x])}
;[o]
                       -20
 -0.318729 + 2.03288 10    I          -1. x
{--------------------------- + 2.71828      + 
             1.46557 x
      2.71828
 
                           -21
   (-0.0377346 + 8.47033 10    I) x
   -------------------------------- + 
                  1.46557 x
           2.71828
 
                                   0.232786 x
   (-0.340635 + 0.660088 I) 2.71828           
 
    (Cos[0.792552 x] - 1. I Sin[0.792552 x]) + 
 
                                   0.232786 x
   (-0.158552 - 0.049372 I) 2.71828           x 
 
    (Cos[0.792552 x] - 1. I Sin[0.792552 x]) + 
 
                                   0.232786 x
   (-0.340635 - 0.660088 I) 2.71828           
 
    (Cos[0.792552 x] + 1. I Sin[0.792552 x]) + 
 
                                   0.232786 x
   (-0.158552 + 0.049372 I) 2.71828           x 
 
    (Cos[0.792552 x] + 1. I Sin[0.792552 x])}
:[font = input; initialization; preserveAspect]
*)
Problems on Differential Operators
(*
:[font = input; initialization; preserveAspect]
*)
Problem 1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
DSolve[{17/32s''[t]-(3-5)s[t]==0,s[0]==-12,s'[0]==0},s[t],t]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{-1/3 + 2^(1/3)/(3*(-29 + 3*93^(1/2))^(1/3)) + 
 
       (-29 + 3*93^(1/2))^(1/3)/(3*2^(1/3)), 
 
      -1/3 + 2^(1/3)/(3*(-29 + 3*93^(1/2))^(1/3)) + 
 
       (-29 + 3*93^(1/2))^(1/3)/(3*2^(1/3)), 
 
      -1/3 - (1 + I*3^(1/2))/
 
        (3*2^(2/3)*(-29 + 3*93^(1/2))^(1/3)) - 
 
       ((1 - I*3^(1/2))*(-29 + 3*93^(1/2))^(1/3))/(6*2^(1/3))
 
       , -1/3 - (1 + I*3^(1/2))/
 
        (3*2^(2/3)*(-29 + 3*93^(1/2))^(1/3)) - 
 
       ((1 - I*3^(1/2))*(-29 + 3*93^(1/2))^(1/3))/(6*2^(1/3))
 
       , -1/3 - (1 - I*3^(1/2))/
 
        (3*2^(2/3)*(-29 + 3*93^(1/2))^(1/3)) - 
 
       ((1 + I*3^(1/2))*(-29 + 3*93^(1/2))^(1/3))/(6*2^(1/3))
 
       , -1/3 - (1 - I*3^(1/2))/
 
        (3*2^(2/3)*(-29 + 3*93^(1/2))^(1/3)) - 
 
       ((1 + I*3^(1/2))*(-29 + 3*93^(1/2))^(1/3))/(6*2^(1/3))
 
       }[t] -> 
 
    -6*E^((-8*I*t)/17^(1/2)) - 6*E^((8*I*t)/17^(1/2))}}
;[o]
                    1/3                               1/3
     1             2                (-29 + 3 Sqrt[93])
{{{-(-) + ----------------------- + ---------------------, 
     3                        1/3             1/3
          3 (-29 + 3 Sqrt[93])             3 2
 
                       1/3
        1             2
      -(-) + ----------------------- + 
        3                        1/3
             3 (-29 + 3 Sqrt[93])
 
                         1/3
       (-29 + 3 Sqrt[93])
       ---------------------, 
                 1/3
              3 2
 
        1           1 + I Sqrt[3]
      -(-) - ---------------------------- - 
        3       2/3                   1/3
             3 2    (-29 + 3 Sqrt[93])
 
                                         1/3
       (1 - I Sqrt[3]) (-29 + 3 Sqrt[93])
       -------------------------------------, 
                         1/3
                      6 2
 
        1           1 + I Sqrt[3]
      -(-) - ---------------------------- - 
        3       2/3                   1/3
             3 2    (-29 + 3 Sqrt[93])
 
                                         1/3
       (1 - I Sqrt[3]) (-29 + 3 Sqrt[93])
       -------------------------------------, 
                         1/3
                      6 2
 
        1           1 - I Sqrt[3]
      -(-) - ---------------------------- - 
        3       2/3                   1/3
             3 2    (-29 + 3 Sqrt[93])
 
                                         1/3
       (1 + I Sqrt[3]) (-29 + 3 Sqrt[93])
       -------------------------------------, 
                         1/3
                      6 2
 
        1           1 - I Sqrt[3]
      -(-) - ---------------------------- - 
        3       2/3                   1/3
             3 2    (-29 + 3 Sqrt[93])
 
                                         1/3
       (1 + I Sqrt[3]) (-29 + 3 Sqrt[93])
       -------------------------------------}[t] -> 
                         1/3
                      6 2
 
        (-8 I t)/Sqrt[17]      (8 I t)/Sqrt[17]
    -6 E                  - 6 E                }}
:[font = input; initialization; preserveAspect; startGroup]
*)
Plot[-12Cos[8t/Sqrt[17]],{t,0,6}]
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Graphics["<<>>"]
;[o]
-Graphics-
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*)
Problem 2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
DSolve[{17/32s''[t]-(3-5)s[t]+s'[t]==0,s[0]==-12,s'[0]==0},s[t],t]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{{-1/3 + 2^(1/3)/(3*(-29 + 3*93^(1/2))^(1/3)) + 
 
       (-29 + 3*93^(1/2))^(1/3)/(3*2^(1/3)), 
 
      -1/3 + 2^(1/3)/(3*(-29 + 3*93^(1/2))^(1/3)) + 
 
       (-29 + 3*93^(1/2))^(1/3)/(3*2^(1/3)), 
 
      -1/3 - (1 + I*3^(1/2))/
 
        (3*2^(2/3)*(-29 + 3*93^(1/2))^(1/3)) - 
 
       ((1 - I*3^(1/2))*(-29 + 3*93^(1/2))^(1/3))/(6*2^(1/3))
 
       , -1/3 - (1 + I*3^(1/2))/
 
        (3*2^(2/3)*(-29 + 3*93^(1/2))^(1/3)) - 
 
       ((1 - I*3^(1/2))*(-29 + 3*93^(1/2))^(1/3))/(6*2^(1/3))
 
       , -1/3 - (1 - I*3^(1/2))/
 
        (3*2^(2/3)*(-29 + 3*93^(1/2))^(1/3)) - 
 
       ((1 + I*3^(1/2))*(-29 + 3*93^(1/2))^(1/3))/(6*2^(1/3))
 
       , -1/3 - (1 - I*3^(1/2))/
 
        (3*2^(2/3)*(-29 + 3*93^(1/2))^(1/3)) - 
 
       ((1 + I*3^(1/2))*(-29 + 3*93^(1/2))^(1/3))/(6*2^(1/3))
 
       }[t] -> 
 
    (-6 - (12*I)/13^(1/2))*
 
      E^(((-32 - 16*I*13^(1/2))*t)/34) + 
 
     (-6 + (12*I)/13^(1/2))*E^(((-32 + 16*I*13^(1/2))*t)/34)}
 
   }
;[o]
                    1/3                               1/3
     1             2                (-29 + 3 Sqrt[93])
{{{-(-) + ----------------------- + ---------------------, 
     3                        1/3             1/3
          3 (-29 + 3 Sqrt[93])             3 2
 
                       1/3
        1             2
      -(-) + ----------------------- + 
        3                        1/3
             3 (-29 + 3 Sqrt[93])
 
                         1/3
       (-29 + 3 Sqrt[93])
       ---------------------, 
                 1/3
              3 2
 
        1           1 + I Sqrt[3]
      -(-) - ---------------------------- - 
        3       2/3                   1/3
             3 2    (-29 + 3 Sqrt[93])
 
                                         1/3
       (1 - I Sqrt[3]) (-29 + 3 Sqrt[93])
       -------------------------------------, 
                         1/3
                      6 2
 
        1           1 + I Sqrt[3]
      -(-) - ---------------------------- - 
        3       2/3                   1/3
             3 2    (-29 + 3 Sqrt[93])
 
                                         1/3
       (1 - I Sqrt[3]) (-29 + 3 Sqrt[93])
       -------------------------------------, 
                         1/3
                      6 2
 
        1           1 - I Sqrt[3]
      -(-) - ---------------------------- - 
        3       2/3                   1/3
             3 2    (-29 + 3 Sqrt[93])
 
                                         1/3
       (1 + I Sqrt[3]) (-29 + 3 Sqrt[93])
       -------------------------------------, 
                         1/3
                      6 2
 
        1           1 - I Sqrt[3]
      -(-) - ---------------------------- - 
        3       2/3                   1/3
             3 2    (-29 + 3 Sqrt[93])
 
                                         1/3
       (1 + I Sqrt[3]) (-29 + 3 Sqrt[93])
       -------------------------------------}[t] -> 
                         1/3
                      6 2
 
            12 I     ((-32 - 16 I Sqrt[13]) t)/34
    (-6 - --------) E                             + 
          Sqrt[13]
 
             12 I     ((-32 + 16 I Sqrt[13]) t)/34
     (-6 + --------) E                            }}
           Sqrt[13]
:[font = input; initialization; preserveAspect; startGroup]
*)
Plot[E^(-16t/17)(-12Cos[8Sqrt[13]t/17]-24/Sqrt[13]Sin[8Sqrt[13]t/17]), \
                                                               {t,0,6}]
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Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = input; initialization; preserveAspect]
*)
Exercises on Recurrence Relations
(*
:[font = input; initialization; preserveAspect]
*)
<<DiscreteMath`RSolve`
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L=Product[d-i,{i,5}]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-120 + 274*d - 225*d^2 + 85*d^3 - 15*d^4 + d^5
;[o]
                    2       3       4    5
-120 + 274 d - 225 d  + 85 d  - 15 d  + d
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[f]
Do[f[i][n_]:=i^n,{i,5}]
LC[n_]:=Sum[k[i]f[i][n],{i,5}]
LC[n]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
k[1] + 2^n*k[2] + 3^n*k[3] + 4^n*k[4] + 5^n*k[5]
;[o]
        n         n         n         n
k[1] + 2  k[2] + 3  k[3] + 4  k[4] + 5  k[5]
:[font = input; initialization; preserveAspect; startGroup]
*)
Table[LC[n]/.Solve[{Array[LC[#-1]&,5]==IdentityMatrix[5][[i]]}, \
                   {k[1],k[2],k[3],k[4],k[5]}],{i,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{5 - 10*2^n + 10*3^n - 5*4^n + 5^n}, 
 
  {-77/12 + (107*2^n)/6 - (39*3^n)/2 + (61*4^n)/6 - 
 
    (25*5^n)/12}, {71/24 - (59*2^n)/6 + (49*3^n)/4 - 
 
    (41*4^n)/6 + (35*5^n)/24}, 
 
  {-7/12 + (13*2^n)/6 - 3*3^n + (11*4^n)/6 - (5*5^n)/12}, 
 
  {1/24 - 2^n/6 + 3^n/4 - 4^n/6 + 5^n/24}}
;[o]
          n       n      n    n
{{5 - 10 2  + 10 3  - 5 4  + 5 }, 
 
                n       n       n       n
     77    107 2    39 3    61 4    25 5
  {-(--) + ------ - ----- + ----- - -----}, 
     12      6        2       6      12
 
            n       n       n       n
   71   59 2    49 3    41 4    35 5
  {-- - ----- + ----- - ----- + -----}, 
   24     6       4       6      24
 
               n              n      n
     7     13 2       n   11 4    5 5
  {-(--) + ----- - 3 3  + ----- - ----}, 
     12      6              6      12
 
         n    n    n    n
   1    2    3    4    5
  {-- - -- + -- - -- + --}}
   24   6    4    6    24
:[font = input; initialization; preserveAspect]
*)
Check:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Table[%[[i]]/.{n->j-1},{i,5},{j,5}]//MatrixForm
(*
:[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}, {1}, {0}}, {{0}, {0}, {0}, {0}, {1}}}]
;[o]
1   0   0   0   0

0   1   0   0   0

0   0   1   0   0

0   0   0   1   0

0   0   0   0   1
:[font = input; initialization; preserveAspect]
*)
Exercise 2
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
s[n]/.RSolve[{s[n+1]-s[n]==(n+1)^2,s[0]==0},s[n],n]//Factor
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{(n*(1 + n)*(1 + 2*n))/6}
;[o]
 n (1 + n) (1 + 2 n)
{-------------------}
          6
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
s[n]/.RSolve[{s[n+1]-s[n]==(n+1)^3,s[0]==0},s[n],n]//Factor
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{(n^2*(1 + n)^2)/4}
;[o]
  2        2
 n  (1 + n)
{-----------}
      4
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
s[n]/.RSolve[{s[n+1]-s[n]==(n+1)^4,s[0]==0},s[n],n]//Factor
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{((-1 - n)*n*(1 + 2*n)*(1 - 3*n - 3*n^2))/30}
;[o]
                                    2
 (-1 - n) n (1 + 2 n) (1 - 3 n - 3 n )
{-------------------------------------}
                  30
:[font = input; initialization; preserveAspect]
*)
d)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
s[n]/.RSolve[{s[n+1]-s[n]==(n+1)^5,s[0]==0},s[n],n]//Factor
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{(n^2*(1 + n)^2*(-1 + 2*n + 2*n^2))/12}
;[o]
  2        2                2
 n  (1 + n)  (-1 + 2 n + 2 n )
{-----------------------------}
              12
:[font = input; initialization; preserveAspect]
*)
Exercise 3
(*
:[font = input; initialization; preserveAspect]
*)
We can conjecture that sums for even n's contain the factors 
n(1+n), whereis sums for odd n's contain (n^2)(1+n)^2.
Verification:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
s[n]/.RSolve[{s[n+1]-s[n]==(n+1)^10,s[0]==0},s[n],n]//Factor
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{(n*(1 + n)*(1 + 2*n)*(-1 + n + n^2)*
 
     (-5 + 10*n + 3*n^2 - 11*n^3 + 2*n^4 + 9*n^5 + 3*n^6))/66
 
   }
;[o]
                                 2
{(n (1 + n) (1 + 2 n) (-1 + n + n ) 
 
                     2       3      4      5      6
     (-5 + 10 n + 3 n  - 11 n  + 2 n  + 9 n  + 3 n )) / 66}
:[font = input; initialization; preserveAspect; startGroup]
*)
s[n]/.RSolve[{s[n+1]-s[n]==(n+1)^15,s[0]==0},s[n],n]//Factor
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{(n^2*(1 + n)^2*(420 - 840*n - 122*n^2 + 1084*n^3 - 
 
       226*n^4 - 632*n^5 + 203*n^6 + 226*n^7 - 83*n^8 - 
 
       60*n^9 + 21*n^10 + 18*n^11 + 3*n^12))/48}
;[o]
   2        2                     2         3        4
{(n  (1 + n)  (420 - 840 n - 122 n  + 1084 n  - 226 n  - 
 
            5        6        7       8       9       10
       632 n  + 203 n  + 226 n  - 83 n  - 60 n  + 21 n   + 
 
           11      12
       18 n   + 3 n  )) / 48}
:[font = input; initialization; preserveAspect]
*)
Exercise 4
(*
:[font = input; initialization; preserveAspect]
*)
Clear[n,s,L,F,d]
d[s_]:=s/.{n->n+1}
F[s_]:=d[s]-2s
L[s_]:=F[F[F[s]]]
(*
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L[2^n]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect; startGroup]
*)
L[n 2^n]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect; startGroup]
*)
L[n^2 2^n]//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
K=Table[{2^(n-1),(n-1)2^(n-1),(n-1)^2 2^(n-1)},{n,3}];
K//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, 0}, {2, 2, 2}, {4, 8, 16}}]
;[o]
1    0    0

2    2    2

4    8    16
:[font = input; initialization; preserveAspect; startGroup]
*)
Det[K]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
16
;[o]
16
:[font = input; initialization; preserveAspect]
*)
We can now conclude that the solutions in part (a) are
a basis for the kernel of L, taking also into account the
fact that the kernel has dimension 3. 
(*
:[font = input; initialization; preserveAspect]
*)
c)
We conjecture that (n^i)(r^n) , with 0 <= i < f , will be
solutions.
(*
:[font = input; initialization; preserveAspect]
*)
L[s_]:=F[F[F[F[s]]]]
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[L[n^i 2^n]//Expand],{i,0,3}]
(*
:[font = print; inactive; preserveAspect; endGroup]
0
0
0
0
:[font = input; initialization; preserveAspect]
*)
Exercise 5
(*
:[font = input; initialization; preserveAspect]
*)
F[s_]:=d[s]+2s
L[s_]:=F[F[s]]+(3^2+2^2)s
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
RSolve[L[s[n]]==0,s[n],n]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{s[n] -> (17/13)^(1/2)*17^(n/2)*
 
      Cos[n*Pi - n*ArcTan[13^(1/2)/2] + 
 
        ArcTan[2*(9/(17*13^(1/2)) - (2*13^(1/2))/17)]]*s[0] \
 
      + (17/13)^(1/2)*17^(-1/2 + n/2)*
 
      Cos[ArcTan[2/13^(1/2)] + 
 
        (-1 + n)*(Pi - ArcTan[13^(1/2)/2])]*If[n >= 1, 1, 0]*
 
      s[1]}}
;[o]
               17    n/2
{{s[n] -> Sqrt[--] 17    
               13
 
                          Sqrt[13]
      Cos[n Pi - n ArcTan[--------] + 
                             2
 
                       9        2 Sqrt[13]
        ArcTan[2 (----------- - ----------)]] s[0] + 
                  17 Sqrt[13]       17
 
          17    -(1/2) + n/2
     Sqrt[--] 17             
          13
 
                    2
      Cos[ArcTan[--------] + 
                 Sqrt[13]
 
                              Sqrt[13]
        (-1 + n) (Pi - ArcTan[--------])] If[n >= 1, 1, 0] 
                                 2
 
      s[1]}}
:[font = input; initialization; preserveAspect]
*)
Problems on Recurrence Relations
(*
:[font = input; initialization; preserveAspect]
*)
Problem 1
a)
(*
:[font = input; initialization; preserveAspect]
*)
One can imagine a recurrent scheme by which, in order to move
i+1 disks once we need M[i+1]=2M[i]+1 moves, since we have to
move the top n, then move the disk # n+1, then move back the
top n.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M[i]/.RSolve[{M[i+1]==2M[i]+1,M[0]==0},M[i],i]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-1 + 2^i}
;[o]
       i
{-1 + 2 }
:[font = input; initialization; preserveAspect]
*)
To achieve our goal, we'll need:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
S=Sum[1+%[[1]],{i,0,n-1}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
Sum[2^i, {i, 0, -1 + n}]
;[o]
     i
Sum[2 , {i, 0, -1 + n}]
:[font = input; initialization; preserveAspect]
*)
i.e.,
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
S=-1+2^n
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-1 + 2^n
;[o]
      n
-1 + 2
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
(S/.{n->64})/((365+1/4)*24*60*60)
(*
:[font = output; output; inactive; preserveAspect; endGroup]
1229782938247303441/2103840
;[o]
1229782938247303441
-------------------
      2103840
:[font = input; initialization; preserveAspect; startGroup]
*)
%//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
5.845420460906263979*10^11
;[o]
          11
5.84542 10
:[font = input; initialization; preserveAspect]
*)
Problem 2
a)
(*
:[font = input; initialization; preserveAspect]
*)
In this case, by the same type of reasoning as before,
if Q[n] is the number of moves required to move n disks from
an end pole to the center and P[n] from the center to an end
pole, we have: P[n+1]=2P[n]+1+Q[n] and Q[n+1]=2Q[n]+P[n]+1.
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
RSolve[{P[n+1]==2P[n]+1+Q[n],Q[n+1]==2Q[n]+P[n]+1,Q[0]==0,P[0]==0}, \
       {P[n],Q[n]},n]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{P[n] -> -1/2 + 3^n/2, Q[n] -> -1/2 + 3^n/2}}
;[o]
                  n                  n
            1    3             1    3
{{P[n] -> -(-) + --, Q[n] -> -(-) + --}}
            2    2             2    2
:[font = input; initialization; preserveAspect]
*)
i.e.,
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Q=(-1+3^n)/2
(*
:[font = output; output; inactive; preserveAspect; endGroup]
(-1 + 3^n)/2
;[o]
      n
-1 + 3
-------
   2
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
(Q/.{n->64})/((365+1/4)*24*60*60)
(*
:[font = output; output; inactive; preserveAspect; endGroup]
10730261938414101514555778404/197235
;[o]
10730261938414101514555778404
-----------------------------
           197235
:[font = input; initialization; preserveAspect; startGroup]
*)
%//N
(*
:[font = output; output; inactive; preserveAspect; endGroup]
5.440343721151976837*10^22
;[o]
          22
5.44034 10
:[font = input; initialization; preserveAspect]
*)
Problem 3
a)
(*
:[font = input; initialization; preserveAspect]
*)
If we denote with "r" the one-month old pairs of rabbits and 
with "R" the pairs of grown-ups, we have: R[n+1]=R[n]+r[n] 
and r[n]=R[n-1], and therefore R[n+1]=R[n]+R[n-1].
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
2R[n]/.RSolve[{R[n+1]==R[n]+R[n-1],R[0]==n0,R[1]==n0},R[n],n] \
    //Factor
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{((-(1/2 - 5^(1/2)/2)^n + 5^(1/2)*(1/2 - 5^(1/2)/2)^n + 
 
       (1/2 + 5^(1/2)/2)^n + 5^(1/2)*(1/2 + 5^(1/2)/2)^n)*n0*
 
     If[n >= 0, 1, 0])/5^(1/2)}
;[o]
     1   Sqrt[5] n            1   Sqrt[5] n
{((-(- - -------)  + Sqrt[5] (- - -------)  + 
     2      2                 2      2
 
        1   Sqrt[5] n            1   Sqrt[5] n
       (- + -------)  + Sqrt[5] (- + -------) ) n0 
        2      2                 2      2
 
     If[n >= 0, 1, 0]) / Sqrt[5]}
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
%[[1]]/.{n0->1}/.{n->12*10}//Expand
(*
:[font = output; output; inactive; preserveAspect; endGroup]
17340014797015897316103842
;[o]
17340014797015897316103842
^*)