Mathematics

A differential equation is homogeneous when you can rewrite it so that $y$ and $x$ appear only as the ratio $v = y/x$. The key move here is to rearrange the equation into the form $\frac{dy}{dx} = f(y/x)$, then substitute $y = vx$ to transform it into a separable equation in $v$ and $x$. Once separated, integrate both sides and substitute back to recover $y$.

A differential equation is homogeneous when you can rewrite it so that $y$ and $x$ appear only as the ratio $v = y/x$. The key move here is to rearrange the equation into the form $\frac{dy}{dx} = f(y/x)$, then substitute $y = vx$ to transform it into a separable equation in $v$ and $x$. Once separated, integrate both sides and substitute back to recover $y$.