Differential equations basic problem

Question

I know this is a basic Physics problems but somehow I can't solve it. We have the differential equation: $2x''x^2 - 4 x^2x' - 2 x^3 = 0$

We have to conclude that the system:

$x' = y $

$y' = 2y + x$

..is equivalent to the differential equation. How can I do it?

Thanks in advance!

Answer 1

If $x \neq 0$, then solving for $x^{\prime\prime}$ we get $$ x^{\prime\prime} = 2x^{\prime} + x $$ Now let $y = x^{\prime}$. Then $$ y^{\prime} = x^{\prime\prime} = 2x^{\prime} + x = 2y + x $$