Reduction from second oder to first order

Question

I wanted to ask how you can reduce an equation of the form y'' + my + y^n=0 to first order. Any resources to figure this out or direction could be helpful.

Answer 1

$$ y'' + my + y^n=0 $$ Multiply by $y'$: $$ y'y'' + myy' + y'y^n=0 $$ $$ \dfrac 12 (y'^2)' + \dfrac m2(y^2)' + \dfrac 1 {n+1}(y^{n+1})'=0 $$ Integrate to reduce the order:

$$ y'^2 + my^2 + \dfrac 2 {n+1}y^{n+1}=C $$

Answer 2

The usual way is to define a new variable and make it a system of two equations, so $$y'=z\\z'=-my-y^n$$