Non-linear second order equation

[Leo Klem]

I am not able to find the general integral of the following non-linear 2nd order equation:

y^2 y'' + a y^3 - b = 0

in which:

y = f(x)

0 < a <= 1, is a constant

b > 0 , is a constant.

 

[coomast]

Hello Leo Klem,

Not an easy one. The only method I know that can be used to solve this equation is Lie point transformations. By setting:

$$u=y$$
$$v=y'$$
and
$$w=\frac{dv}{du}=\frac{y''}{y'}$$

You can transform the equation into:

$$v\frac{dv}{du}=\frac{b}{u^2}-au$$

Which has as solution:

$$v^2=-\frac{2b}{u}-au^2+K_1$$

Taking the root and inverting the substitution you end up with:

$$\pm x+K_2=\int \frac{\sqrt{y}dy}{\sqrt{K_1y-2b-ay^3}}$$

Which is not an easy integral. Maybe by trying to find the roots you can solve it, I did not try this. (check the calculations for errors to be sure)

Hope this helps so far,

coomast

 

[g_edgar]

It's an elliptic integral, not elementary.
Maple evaluates it in terms of the elliptic integrals $F$ and $\Pi$, but too complicated to copy here.

 

[Leo Klem]

16 August 2009

Many thanks for the attention paid both by "coomast" and by "g_edgar".
The solution for x given by "ccomast" is just the point where I too had to stop. Unfortunately, I am not in condition to calculate the integral indicated.
Leo Klem