Homogeneous Differential Equations

A first-order equation is homogeneous when it can be written so that $\dfrac{dy}{dx}$ depends on $x$ and $y$ only through their ratio $\dfrac{y}{x}$:

Equivalently, in the form $M(x,y)\,dx + N(x,y)\,dy = 0$, both $M$ and $N$ are homogeneous functions of the same degree (every term has the same total power in $x$ and $y$). Such equations are not directly separable, but a substitution converts them into a separable one.

The substitution. Put

Replacing $y/x$ by $v$ turns the equation into one in $v$ and $x$ that always separates. Solve for $v$, then substitute back $v = y/x$.

Worked illustration. Solve $\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{2xy}$.

The right side depends only on $y/x$, so it is homogeneous. Put $y = vx$:

Then

Separate and integrate:

Hence $\ln|1-v^2| + \ln|x| = -c_1$, so $x(1 - v^2) = k$ for a constant $k$. Restoring $v = y/x$:

$x\left(1 - \frac{y^2}{x^2}\right) = k \;\Longrightarrow\; \frac{x^2 - y^2}{x} = k \;\Longrightarrow\; x^2 - y^2 = kx.$ …