the substitution v = y/x

This is the specific change of variable that cracks a homogeneous equation. The whole reason a homogeneous equation is solvable is that its slope depends only on the ratio y/x; so you give that ratio its own name, v, and rewrite everything in terms of v and x. The new equation is always separable.

Here is the mechanism in plain steps. Set v = y/x, which means y = v x. Differentiate the product with respect to x using the product rule: dy/dx = v + x dv/dx. Substitute both y = v x and this expression for dy/dx into the homogeneous equation dy/dx = f(y/x) = f(v). You get v + x dv/dx = f(v), and the v terms can be gathered: x dv/dx = f(v) - v. This separates into dv/(f(v) - v) = dx/x — pure separation of variables. Integrate both sides, then replace v by y/x to return to the original variables.

The trick is worth internalizing because it is the template for all substitution methods: spot the single combination the equation truly depends on, name it, and the equation collapses to something you already know how to solve. The same instinct turns a Bernoulli equation linear and reduces a Riccati equation once a solution is known.