Homogeneous equations

A homogeneous differential equation is of the form \(\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{f}\left(\frac{y}{x}\right)\).

The total power of \(x\) and \(y\) are equal for every term in the equation.

These are examples of homogeneous differential equations:

1. \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{3x + y}{x - y}\)
2. \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^2 + y^2}{2xy}\)
3. \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^3 + y^3}{xy^2 + yx^2}\)

We can convert homogeneous differential equations into variable separable differential equations, \(\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{f}\left(\frac{y}{x}\right) = \mathrm{f}(v)\), by using the substitution \(y = vx\), where \(v\) is a function of \(x\).

Differentiating \(y = vx\) with respect to \(x\), we get \(\frac{\mathrm{d}y}{\mathrm{d}x} = v + x \frac{\mathrm{d}v}{\mathrm{d}x}\).
Combining both equations, we get

\[v + x \frac{\mathrm{d}v}{\mathrm{d}x} = \mathrm{g}(v)\]

which is a variable separable differential equation.