Approach for non linear ODE with quadratic derivative.

Question

What is the approach to solve the following nonlinear ODE:

$$\ddot{x}(t) + a(\dot{x}(t))^2+b\dot{x}(t)+kx(t) = 0$$

where $a$ is positive when $\dot{x}<0$ and negative otherwise. I am trying to linearise it without success.