Non Homogeneous into Homogeneous differential equations

Question

How can I turn this differential equation into a homogeneous differential equation $$y' = 2x + \frac{8y}{x} + 1$$?

Answer 1

Assuming you are after the solution to the above, here is how I would tackle it: $$ y'-\frac{8}{x}y=2x+1 $$ And mulitplying both sides by $\exp(-8\int\frac{1}{x}dx)=\frac{1}{x^8}$, we get an exact equation $$ \frac{1}{x^8}y'-\frac{1}{x^9}y=\frac{2x+1}{x^8}\Rightarrow(\frac{1}{x^8}y)'=\frac{2x+1}{x^8}\Rightarrow \frac{y}{x^8}=\int\frac{2x+1}{x^8}\\ \Rightarrow \frac{y}{x^8}=\frac{-1}{4x^8}-\frac{1}{x^9}+c\Rightarrow y=cx^8-\frac{1}{4}-1/x $$