1. ## Nonlinear ODE

Here is the equation $\dfrac{dy}{dx}\dfrac{d^2 y}{dx^2}=y\dfrac{d^3 y}{dx^3}$

Any ideas?

2. Two thoughts:

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

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

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

You can integrate then, to obtain

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

You can go from there further, I think.

3. There's also a constant of integration to think about.

4. Indeed. As always. So you've really got

$\ln|y|=\ln|y''|+C_{1}$ so far.

5. Might be a little easier to write ( $y \ne 0$)

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

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

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