## 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.

The above version of our differential equation has been separated.

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

INTERESTING.