Re: symbolic solution of ODE

• To: mathgroup at smc.vnet.net
• Subject: [mg71510] Re: symbolic solution of ODE
• From: "ben" <benjamin.friedrich at gmail.com>
• Date: Tue, 21 Nov 2006 07:05:07 -0500 (EST)

```Hi

I do not know how to teach Mathematica to simplify its result for the cast
t^2<k.
However you can read off the solution yourself:
The two double-sqrt-expressions appearing in the exponent are complex
conjugate; just introduce new notation a+bi and a-bi and let Mathematica
replace the old by the
new notation. Then you "see" the solution immeadiatly

C[5] Exp[a z] Cos[b z + C[6]]+C[7] Exp[-a z] Cos[b z + C[8]]

Bye Ben

the replacement-rule:

In[33]:=
\!\(\(w[z] /.
sol[\([1]\)]\) /. {\@\(t - \@\(\(-k\) + t\^2\)\) \[Rule]
a + b\ I, \@\(t + \@\(\(-k\) + t\^2\)\) \[Rule] a - b\ I}\)

Out[33]=
\!\(\[ExponentialE]\^\(\((a + \[ImaginaryI]\ b)\)\ z\)\ C[
1] + \[ExponentialE]\^\(\(-\((a + \[ImaginaryI]\ b)\)\)\ z\)\
C[
2] + \[ExponentialE]\^\(\((a - \[ImaginaryI]\ b)\)\ z\)\ C[
3] + \[ExponentialE]\^\(\(-\((a - \[ImaginaryI]\ b)\)\)\ z\)\
C[4]\)

C[5] Exp[a z] Cos[b z + C[6]]+C[7] Exp[-a z] Cos[b z + C[8]]

visbuga at purdue.edu schrieb:

> 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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