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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.


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