solve a problem of second order nonlinear ODE I want to solve a second order nonlinear ODE problem $y^2 \frac{d^2y}{dx^2}+m(\frac{dy}{dx})^3+ny(\frac{dy}{dx})^2-ky=0$. 
 A: You could write $x$ as a function of $y$.  Using things like $dy/dx=1/(dx/dy)$, this will turn the second-order DE into a first-order DE.  That happens because $x$ does not appear as an independent parameter, but only in the derivatives.
Or, you might find a solution for small $x$ and small $y$ as $y(x)=x^\alpha g(x)$, and $g(x)$ has an ordinary Taylor series.  Choose $\alpha$ so that, when $x$ and $y$ are small,
two of the four terms are as big as each other and the other two terms are smaller.
Solve the equation with just two large terms; then the two small terms just perturb
that basic solution.
A: Let $u=\dfrac{dy}{dx}$ ,
Then $\dfrac{d^2y}{dx^2}=\dfrac{du}{dx}=\dfrac{du}{dy}\dfrac{dy}{dx}=u\dfrac{du}{dy}$
$\therefore y^2u\dfrac{du}{dy}+mu^3+nyu^2-ky=0$
$u\dfrac{du}{dy}=-\dfrac{mu^3}{y^2}-\dfrac{nu^2}{y}+\dfrac{k}{y}$
This gives a separable ODE when $m=0$ .
When $m\neq0$ ,
Let $u=\dfrac{1}{v}$ ,
Then $\dfrac{du}{dy}=-\dfrac{1}{v^2}\dfrac{dv}{dy}$
$\therefore-\dfrac{1}{v^3}\dfrac{dv}{dy}=-\dfrac{m}{y^2v^3}-\dfrac{n}{yv^2}+\dfrac{k}{y}$
$\dfrac{dv}{dy}=-\dfrac{kv^3}{y}+\dfrac{nv}{y}+\dfrac{m}{y^2}$
This gives a linear ODE when $k=0$ , an Abel equation of the first kind when $k\neq0$ . When $k\neq0$ , http://www.hindawi.com/journals/ijmms/2011/387429/#sec2 claims that it has analytical method to solve it.
