Solution by Separation of Variables

The simplest first-order, first-degree differential equations to solve are those whose right-hand side, when written as $\dfrac{dy}{dx} = h(x,y)$, factorises as a function of $x$ alone times a function of $y$ alone:

Such an equation is said to have variables separable.

Method. Provided $g(y) \neq 0$, divide both sides by $g(y)$ and multiply by $dx$ to collect every $y$-term (together with $dy$) on one side and every $x$-term (together with $dx$) on the other:

Both sides are now ordinary single-variable expressions, so each can be integrated independently:

where a single arbitrary constant $C$ is added (the two separate constants of integration are absorbed into one, since only their difference matters). Carrying out both integrals and simplifying gives the general solution, usually as an implicit relation between $x$ and $y$; if possible, this is then solved explicitly for $y$.

Worked derivation. To solve $\dfrac{dy}{dx} = \dfrac{x}{y}$ ($y \neq 0$): separate, $y\,dy = x\,dx$; integrate both sides, $\dfrac{y^2}{2} = \dfrac{x^2}{2} + C_1$; multiply by $2$ and rename the constant, $y^2 - x^2 = C$. Substituting back confirms this satisfies the original equation for every value of $C$: differentiating $y^2 - x^2 = C$ implicitly gives $2y\,y' - 2x = 0$, i.e. $y' = x/y$, exactly the equation started with. …