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MathGroup Archive 2006

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Re: symbolic solution of ODE

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71620] Re: symbolic solution of ODE
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Sat, 25 Nov 2006 05:36:59 -0500 (EST)
  • References: <ejtdfo$iva$1@smc.vnet.net><ejuqn8$i0b$1@smc.vnet.net>

Further on my previous post...

Copy paste the following in a notebook.

sol = DSolve[Derivative[4][w][z] - 2*t*Derivative[2][w][z] + k*w[z] ==
0, w[z], z, GeneratedParameters -> (Subscript[c, #1] & )]
derfun = Flatten[(D[sol, {z, #1}] & ) /@ Range[0, 4]];
FullSimplify[Derivative[4][w][z] - 2*t*Derivative[2][w][z] + k*w[z] ==
0 //. %]
sol = w[z] /. FullSimplify[sol /. -k + t^2 -> -m^2, m > 0][[1]]

eq1 = Sqrt[(-I)*m + t] == a + I*b
ComplexExpand[(#1^2 & ) /@ eq1]
(ComplexExpand[Re[#1]] & ) /@ %
(ComplexExpand[Im[#1]] & ) /@ %%
FullSimplify[Solve[% && %%, {a, b}]]
%[[1]]
f = a + I*b /. %
FullSimplify[f^2 == (-I)*m + t]

eq2 = Sqrt[I*m + t] == c + I*d
ComplexExpand[(#1^2 & ) /@ eq2]
(ComplexExpand[Re[#1]] & ) /@ %
(ComplexExpand[Im[#1]] & ) /@ %%
FullSimplify[Solve[% && %%, {c, d}]]
%[[2]]
g = c + I*d /. %
FullSimplify[g^2 == I*m + t]

dat = {Random[], Random[]}
Thread[{m, t} -> dat]
N[Sqrt[I*m + t] - Sqrt[(-I)*m + t] /. %]
f - g /. %%

sol2 = sol /. {Sqrt[(-I)*m + t] -> f, Sqrt[I*m + t] -> g}

sol3 = sol2 /. {m/(Sqrt[2]*Sqrt[-t - Sqrt[m^2 + t^2]]) -> k, Sqrt[-t -
Sqrt[m^2 + t^2]]/Sqrt[2] -> n}

Regards
Dimitris


dimitris wrote:
> Hello.
>
> First the solution of your ODE.
>
> sol = DSolve[Derivative[4][w][z] - 2*t*Derivative[2][w][z] + k*w[z] ==
> 0, w[z], z, GeneratedParameters -> (Subscript[c, #1] & )]
>
> Let's check it.
>
> derfun = Flatten[(D[sol, {z, #1}] & ) /@ Range[0, 4]];
>
> FullSimplify[Derivative[4][w][z] - 2*t*Derivative[2][w][z] + k*w[z] ==
> 0 //. %]
> True
>
> >...imposing the condition "t^2 < k"...
>
> Did you mean something like the following?
>
> w[z] /. FullSimplify[sol /. -k + t^2 -> -m^2, m > 0][[1]]
>
> Dimitris
>
> visbuga at purdue.edu wrote:
> > Hello,
> >
> > I like to solve the ODE
> > DSolve[w''''[z] - 2*t*w''[z] + k*w[z] == 0, w[z], z]
> > by imposing the condition "t^2 < k" on constants in the eqn.In fact, solution
> > of this homogenous ODE is known, but I want to see the result of mathematica
> > to use in a complex eqn. Thank you.
> > 
> > VV


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