## Homogeneous First-order Equations

There are two ways to view "homogeneous" first-order ordinary differential equations.

1. \(\frac{dy}{dx}=f(x,y)\) is a general first order ordinary differential equation. If the function on the right, \(f(x,y)\), can be rewritten as a function of \(\frac{y}{x}\), e.g. \(f(x,y)=\frac{y-2x}{x}\) can be rewritten as \(g\left(\frac{y}{x}\right)=\frac{y}{x}-2\). If we let \(v=\frac{y}{x}\), then we would have \(g(v)=v-2\).

In general, the substitution, \(v=\frac{y}{x}\) or \(y=vx\), will lead to a separable differential equation.

Since \(y=vx\) , differentiating both sides gives \(\frac{dy}{dx}=\frac{dv}{dx}x+v\). This new version of \(\frac{dy}{dx}\) becomes our new left side and \(g(v)\) becomes the right.

\[\frac{dv}{dx}x+v=g(v)\]

\[\frac{dv}{dx}x=g(v)-v\]

\[\frac{1}{g(v)-v}dv=\frac{1}{x}dx\]

2. First-order linear equations, \(\displaystyle \frac{dy}{dx}+P(x)y=Q(x)\), are also separable if \(Q(x)\equiv 0\).

\[\frac{dy}{dx}+P(x)y=0\]

\[\frac{dy}{dx}=-P(x)y\]

\[\frac{1}{y}dy=P(x)dx\]