<s>[INST] Non linear effect in differential equation Suppose I want to study the non linear ODE
$\frac{d^2}{dx^2}f(x) + a f^2(x) + b f(x) + c=0$
I know the solution for $a=0$ and I know $a$ is a small parameter. How can I study the effect of the non linear term?
 [/INST] Multiply by $f'$ to yield a integrable equation. After it depends on $f$ if you can do the last integral.
$\textbf{Edit:}$
$$
f'' f' + \left[af^2+bf + c\right]f' = f'\dfrac{d}{dx}f' + \frac{d}{dx}\left[\frac{1}{3}af^3 + \frac{1}{2}bf^2+cf\right] = 0
$$
now we can re-write the last but one equation as
$$
\dfrac{d}{dx}\left[\frac{f'^2}{2} + \frac{1}{3}af^3 + \frac{1}{2}bf^2+cf\right] = 0
$$
or 
$$
\frac{f'^2}{2} + \frac{1}{3}af^3 + \frac{1}{2}bf^2+cf = \text{Constant} = \lambda
$$
then we have a separable equation
$$
\dfrac{df}{dx} = \pm \sqrt{\lambda -\left(\frac{1}{3}af^3 + \frac{1}{2}bf^2+cf \right)}
$$
now to solve this (maybe) heinous equation..we require the $f(x)$ to play nice ;).
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