For which value of $n$ is $(x^{n}y + y^{4})\,dx = x^{2}y^{2}\,dy$ a homogeneous differential equation?

Explanation

1. 01

   $\frac{dy}{dx} = \frac{x^{n}y + y^{4}}{x^{2}y^{2}}$; both $x^{2}y^{2}$ and $y^{4}$ have degree 4.

2. 02

   So $x^{n}y$ needs degree 4 as well: $n + 1 = 4$, $n = 3$.