I don't know how this energy function (screenshot below) comes from the oscillator equation. I know you can get it from $E =\frac{{\dot x}^2}{2} - \int \ddot x (x)dx$, which is conservative (meaning $\dot E = 0$), which is only point of an "energy" function in the book til this page (as supported by the second screenshot). But, how do we know this energy function has anything to do with this diff eq?
And, why is it saying that after a cycle the change in E is zero. That would mean it actually is a conserved quantity. But, the time derivative of E is ${\dot x}^4$, so it is not a conserved quantity.
(Also, any editing tips would be appreciated.)
Second screenshot:
