Here is the equation $\displaystyle \dfrac{dy}{dx}\dfrac{d^2 y}{dx^2}=y\dfrac{d^3 y}{dx^3}$
Any ideas?
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.
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) $\displaystyle c_1 = - \omega^2.$