Q.A homogeneous differential equation of the form $\frac{dx}{dy} = f\left(\frac{x}{y}\right)$ can be solved by making substitution.

Use $x = vy$.

When a differential equation has the homogeneous form $\dfrac{dx}{dy} = f\!\left(\dfrac{x}{y}\right)$, the right side depends only on the ratio $\dfrac{x}{y}$. The standard technique is to set that ratio equal to a new variable:

$x = vy \quad\Rightarrow\quad \frac{x}{y} = v, \qquad \frac{dx}{dy} = v + y\frac{dv}{dy}.$

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