Homogeneous Functions | Equations of Order One

If the function f(x, y) remains unchanged after replacing x by kx and y by ky, where k is a constant term, then f(x, y) is called a homogeneous function. A differential equation

$M \, dx + N \, dy = 0 \,\, \to \,\,$ Equation (1)

is homogeneous in x and y if M and N are homogeneous functions of the same degree in x and y.

To solve for Equation (1) let

$y = vx \,\, \text{and} \,\, dy = v \, dx + x\, dv$


$x = vy \,\, \text{and} \,\, dx = v \, dy + y\, dv$

The substitution above will lead to variables separable differential equation.