Trying to obtain an explicit solution of a second order ODE
Subject: Trying to obtain an explicit solution of a second order ODE
From: Rupert Swarbrick
Date: Wed, 17 Apr 2013 13:51:25 +0100
deltaquattro at gmail.com writes:
> Hi, all,
>
> I have the following ODE:
>
> assume(C>0);
>
> de: 'diff(y,x,2)=-C*'diff(y,x)^2/y;
>
> ode2 solves it easily:
>
> ode2(de,y,x);
>
> (%o5) (y*%e^(log(y)*C))/(%k1*C+%k1)=x+%k2
>
> I would like to get an explicit solution, in order to check my own explicit
> solution found by hand. How can I do that? Thanks,
I think you want the "ic2" function. For example here's the general
solution to your equation, parameterised by the initial value y0 and
derivative dy0.
(%i12) assume(C>0)$
(%i13) de: 'diff(y,x,2)=-C*'diff(y,x)^2/y$
(%i14) soln: ode2(de,y,x)$
(%i15) ic2(soln, x=0, y=y0, 'diff(y,x)=dy0);
log(y) C
y %e y0
(%o15) ------------------- = ----------- + x
C C dy0 C + dy0
dy0 y0 C + dy0 y0
(%i16) first (solve(%, y));
C C + 1 - log(y) C
(%o16) y = (dy0 x y0 (C + 1) + y0 ) %e
Rupert
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