As the header says: I want to solve the differential equation $ m \ddot x +\alpha x + \beta x^3 = 0$, with initial conditions $x(0) = -x_0$, $\dot x(0)=0$. It comes up in the solution to the equations of motion of the so-called undamped duffing oscillator, an undamped oscillator driven by nonlinear force $-\alpha x - \beta x^3$. My research has shown that the solution involves elliptic integrals, which I have never seen before.
Here is my approach so far: I use conservation of energy. The total Energy should be $E_{tot} = T + V = \frac{1}{2}m \dot x^2 +\alpha \frac{x^2}{2}+ \beta \frac{x^4}{4} = E_{t= 0} = \alpha \frac{x_0^2}{2}+ \beta \frac{x_0^4}{4} $. Now I would transfrom this equation into $\dot x = \sqrt{\frac{2}{m}(\alpha \frac{x^2-x_0^2}{2}+ \beta \frac{x_0^4-x^4}{4})}$ and then obtain by seperation of variables $dt = \frac{dx}{\sqrt{\frac{2}{m}(\alpha \frac{x^2-x_0^2}{2}+ \beta \frac{x_0^4-x^4}{4})}}$. By integrating both sides, I'm guessing the integral on the right $$\int \frac{dx}{\sqrt{\frac{2}{m}(\alpha \frac{x^2-x_0^2}{2}+ \beta \frac{x_0^4-x^4}{4})}}$$ is the elliptic integral.
Is my reasoning correct so far? And if yes, how do you solve (analytically) the integral?