Solving by the Method of Variable Separation

The variable-separable method solves a differential equation of the special form $\dfrac{dy}{dx} = g(x)\,h(y)$, where the right side can be written as a function of $x$ alone multiplied (or divided) by a function of $y$ alone. This chapter restricts the method to algebraic functions of $x$ and $y$ — rational expressions built from $x$, $y$, and whole-number powers — and never introduces the linear, exact, or homogeneous differential-equation methods that a fuller calculus course would cover next; any equation that cannot be split this cleanly into an “$x$-only” factor and a “$y$-only” factor is outside this chapter's scope.

Method:

1. Check that the equation genuinely separates: rearrange so every term involving $y$ (including $dy$) sits on one side and every term involving $x$ (including $dx$) sits on the other — $\dfrac{1}{h(y)}\,dy = g(x)\,dx$.
2. Integrate both sides independently, using the standard integral formulae already met in the Integration chapter (e.g. $\int x^n\,dx = \dfrac{x^{n+1}}{n+1}+C$, $\int \dfrac{1}{x}\,dx = \ln|x|+C$).
3. Combine the two constants of integration that arise (one from each side) into a single arbitrary constant, since both are arbitrary real numbers — writing two separate constants would be redundant.
4. Simplify the resulting relation between $x$ and $y$ into as clean a form as possible — this is the general solution.

Illustration: solve $\dfrac{dy}{dx} = \dfrac{x^2}{y^2}$. Separating, $y^2\,dy = x^2\,dx$. Integrating both sides, $\dfrac{y^3}{3} = \dfrac{x^3}{3} + c$, which simplifies to $y^3 - x^3 = k$ (writing $k=3c$ for the combined constant) — the general solution.