Differential Equations with Variables Separable

9.4.1 Differential Equations with Variables Separable

The Core Idea: When a Derivative Factorises

A first-order, first-degree differential equation has the general form

The difficulty of solving it depends on the structure of $F(x, y)$. The simplest case — treated first — occurs when $F(x, y)$ factorises into a function of $x$ alone times a function of $y$ alone.

If $F(x, y) = g(x) \cdot h(y)$, the differential equation becomes

a variable separable differential equation.

The Separation Procedure

Provided $h(y) \neq 0$, treat $dy$ and $dx$ as algebraic quantities (Leibniz's notation) and separate the variables:

Now integrate both sides:

Writing $H(y)$ for an antiderivative of $\frac{1}{h(y)}$ and $G(x)$ for one of $g(x)$, equation (4) yields

where $C$ is an arbitrary constant of integration. This is the general solution — a whole family of curves, one for each value of $C$. …