Solve $x \ddot {x} + {\dot {x}}^2=0$. I solved a physics problem and I got this equation, but I don't know how to proceed. Could you solve for $x (t)$ this equation:
$x \ddot {x} + {\dot {x}}^2=0$

$$x\ddot{x} + \dot{x}^2=\frac{d}{dt}(x\dot{x})=0\\
  x\frac{dx}{dt}=c_1\\
\int xdx=\int c_1 dt\\
\frac{x^2}{2}=c_1t+c_2\\
x=\sqrt{2c_1t+c_2}$$
which may be rewritten as 
$$x=c_1\sqrt{2t+c_2}$$
