Introduction
Integrating equations comprised of multiple variables is difficult. Separation of Variables is the technique of separating the variables in a differential equation to simplify the integration.
In other words, we take equations of either of the two forms:
\frac{dy}{dx} = \frac{f(x)}{g(y)} \quad \text{or} \quad f(x)dx + g(y)dy = 0
Convert them to the form:
f(x)dx = g(y)dy
And integrate
\int f(x)dx = \int g(y)dy
Examples
Example #1:
\frac{dy}{dx} = kx \\ \text{} \\
dy = kx \, dx \\ \text{} \\
\int dy = \int kx \, dx \\ \text{} \\
y = \frac{kx^2}{2} + C
Example #2:
\frac{dy}{dx} = -\frac{x}{y} \\ \text{} \\
y \, dy = -x \, dx \\ \text{} \\
\int y \, dy = \int -x \, dx \\ \text{} \\
\frac{y^2}{2} = -\frac{x^2}{2} + C \\ \text{} \\
\frac{y^2}{2} + \frac{x^2}{2} = C \\ \text{} \\
y^2 + x^2 = C \\ \text{} \\
(The slope of a circle is
-x/y
.)
Example #3:
\begin{align*}
x^2 \, dx + \sin(y) \, dy &= 0 \\ \text{} \\
x^2 \, dx &= -\sin(y) \, dy \\ \text{} \\
\int x^2 \, dx &= \int -\sin(y) \, dy \\ \text{} \\
\frac{x^3}{3} &= \cos(y) + C \\ \text{} \\
\end{align*}