# Nonlinear ODE

• Jul 23rd 2010, 08:03 AM
fobos3
Nonlinear ODE
Here is the equation $\displaystyle \dfrac{dy}{dx}\dfrac{d^2 y}{dx^2}=y\dfrac{d^3 y}{dx^3}$

Any ideas?
• Jul 23rd 2010, 08:15 AM
Ackbeet
Two thoughts:

1. Any affine function y = mx + b solves the equation trivially.
2. You could try this:

$\displaystyle y'\,y''=y\,y'''$, implies

$\displaystyle \displaystyle{\frac{y'}{y}=\frac{y'''}{y''}}.$

You can integrate then, to obtain

$\displaystyle \ln|y|=\ln|y''|.$

You can go from there further, I think.
• Jul 23rd 2010, 09:34 AM
Jester
There's also a constant of integration to think about.
• Jul 23rd 2010, 09:49 AM
Ackbeet
Indeed. As always. So you've really got

$\displaystyle \ln|y|=\ln|y''|+C_{1}$ so far.
• Jul 23rd 2010, 10:01 AM
Jester
Might be a little easier to write ($\displaystyle y \ne 0$)

$\displaystyle y y''' - y' y'' = 0$ as $\displaystyle \dfrac{d}{dx} \left(\dfrac{y''}{y}\right) = 0$

so $\displaystyle y'' = c_1 y$ then look at cases when

(i) $\displaystyle c_1 = \omega^2$,
(ii) $\displaystyle c_1 = 0,$
(iii) $\displaystyle c_1 = - \omega^2.$