Re: [Q] Differential equation?
- To: mathgroup at smc.vnet.net
- Subject: [mg22640] Re: [Q] Differential equation?
- From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
- Date: Thu, 16 Mar 2000 09:10:55 -0500 (EST)
- Organization: Universitaet Leipzig
- References: <8an28e$1u9@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Hi James,
you can solve this equation with Mathematica
a) reduce it to a second order equation for y[1][t]
deqn2=y[1]''[t] + (a + b - c*t)*y[1]'[t] - (c + a*c*t)*y[1][t] ==0
b) remove the first derivative with
deqn3 = (deqn2 /.
Thread[{#, D[#, t], D[#, t, t]} & /@ (y[1][t] ->
Exp[Integrate[-a - b + c*t, t]/2]*u[t])] //
FullSimplify) /. (g_Power*f_Plus == 0) :> f == 0
gives
(a^2 + 2*c + (b - c*t)^2 + 2*a*(b + c*t))*u[t] - 4 u''[t] == 0
hmm - something hypergeometric fine, let's try DSolve[]
DSolve[deqn3, u[t], t] // FullSimplify
and get
{{u[t] -> (((a - b + c*t)^2)^(3/4)*
(C[2]*Hypergeometric1F1[(3 + Sqrt[c^(-2)]*(2*a*b + c))/
4, 3/2, (Sqrt[c^(-2)]*(a - b + c*t)^2)/2] +
C[1]*HypergeometricU[(3 + Sqrt[c^(-2)]*(2*a*b + c))/4,
3/2, (Sqrt[c^(-2)]*(a - b + c*t)^2)/2]))/
(2*E^((Sqrt[c^(-2)]*(a - b + c*t)^2)/4)*
(-(c^2*(a - b + c*t)^2))^(1/4))}}
It is up to you to revert the transformations.
Hope that helps
Jens
James wrote:
>
> Hi!
>
> I began to use Mathematica, and found out it is great.
> But I happen to have a question during solving differential equtations.
> Here's a problem.
>
> y'_0(t) = -a * y_0(t) + b * y_1(t)
> y'_1(t) = a * y_0(t) + (c*t-b) * y_1(t) --- (*)
> ^
> This can be solvable mathematically, even some tedious work,
> but when I use Mathematica, it can't solve it.
> After some trial and error, I found out that 't' in (*)
> is the problem - problem that mathematica doesn't give an answer,
> it just shows the above equations as an answer.
> So I wonder if this is the limit of Mathematica,
> or is there any way to solve it?
> I sincerely hope there's some way - because my work involves
> a lot of Diffrential Equations.
> Any reply would be appreciated.
>
> James.