       Simple ODE with time-dep BC

• To: mathgroup at smc.vnet.net
• Subject: [mg77933] Simple ODE with time-dep BC
• From: "Apostolos E. A. S. Evangelopoulos" <a.e.a.evangelopoulos at sms.ed.ac.uk>
• Date: Wed, 20 Jun 2007 05:27:35 -0400 (EDT)

```Hello all,

I'm having 2 problems with solving the following ordinary differential equation:

h*h'[t]==-a*h^2+c

Problem 1:
Mathematica doesn't like the form of the above:
DSolve[{h'[t]\[Equal]-a*h[t]+c/h, h\[Equal]0}, h[t], t]
returns
DSolve::dvnoarg : The function h appears with no arguments. More...

If, now, I reduce the above equation to
DSolve[{1/2*u'[t]\[Equal]-a*u[t]+c,u\[Equal]0},u[t],t]
(by substitution of u=h^2)
then Mathematica solves for u[t] without complaints. The thing is, though, I could solve the latter by hand, really, so what's the point? Is Mathematica not supposed to be able to solve the equation in the first form, really (it appears to be non-linear, but intrinsically it's not)?

Problem 2:
I 'd like to impose a time dependent boundary condition, so, instead of h==0, something like h[t]==v (some constant), or, the more complicated, h[t]==t. How do I solve that? I have tried as follows, with the reduced u[t] form:
DSolve[{u'[t]/2\[Equal]-a*u[t]+c,u[t]\[Equal]v^2},u[t],t]
resulting in
DSolve::overdet : The system has fewer dependent variables than equations, so is overdetermined. More...