Solving Second-Order Nonlinear ODE Consider the ODE $\frac{d^2x}{dt^2} = a x + b x^2$, where $a$ and $b$ are some parameters. I wonder how I can solve this ODE?
Any comment/hint is greatly appreciated.

Multiplying both sides by $\frac{dx}{dt}$ you get
$$\frac{1}{2} \frac{d}{dt}\left(\frac{dx}{dt}\right)^2 = \frac{d}{dt}\left(\frac{a}{2} x^2(t) +  \frac{b}{3} x^3(t)\right)$$ and by integration
$$\left(\frac{dx}{dt}(t)\right)^2 = a x^2(t) +  \frac{2b}{3} x^3(t)+ C$$ where $C$ is a real constant. By integration again you get
$$\int_{x_0}^{x(t)} \frac{du}{\sqrt{a u^2 +  \frac{2b}{3} u^3+ C}} = \pm(t-t_0)$$
