Separating $y$'s and $x$'s

Question

I am doing this initial value problem where I have the equation $y' + \frac{3}{x} y=\frac{\cos(x)}{x^3} $. I know how to do these kinds of problems but I am having trouble getting the $x$ to the right side and the $y$ into the left side, any help would be appreciated.

Answer 1

You'll have to treat this like a first order linear (non-separable) differential equation. Multiply both sides by $x^3$. You'll see that the left side looks like the result of a product rule expansion, giving you $$[yx^3]’=\cos(x)$$ Then, you can integrate, to yield $$yx^3=-\sin x + C$$ and divide by $x^3$ to get $$y=-\frac{\sin x}{x^3} + Cx^{-3}$$