Homogeneous Type Differential Equations

A homogeneous type differential equation is a ordinary differential equation that can be expressed in the form

$$\frac{dy}{dx}=f(x,y)$$

which satisfies

$$f(tx,ty)=f(x,y),\text{for any }t\ne 0$$

e.g.

$$\frac{dy}{dx}=\frac{x^2+xy}{x^2+3xy}$$

Solving

Method 1

Let $y=vx$ then,

$$\begin{array}{rl}v& =x\frac{dv}{dx}=f(1,v)\\ x\frac{dv}{dx}=f(1,v)-v& \\ \frac{dv}{f(1,v)-v}& =\frac{dx}{x}\end{array}$$

The result is a separable differential equation.

Example 1

$$\frac{dy}{dx}=\frac{3y^2-x^2}{2xy}$$

Let $y=vx$ then,

$$\begin{array}{rl}v+x\frac{dv}{dx}& =\frac{3v^2{x}^2-x^2}{2x^{2}v}=\frac{3x^2-1}{2v}\\ x\frac{dv}{dx}& =\frac{3v^2-1}{2v}-v=\frac{v^2+1}{2v}\\ \int \frac{2v}{v^2+1}dv& =\int \frac{dx}{x}\end{array}$$

This is now a separable DE.