Separation of Variables

Theorem
Suppose a first order ordinary differential equation can be expressible in this form:
 * $\dfrac {\d y} {\d x} = \map g x \, \map h y$

Then the equation is said to have separable variables, or be separable.

Its general solution is found by solving the integration:
 * $\displaystyle \int \frac {\d y} {\map h y} = \int \map g x \rd x + C$

Proof
Dividing both sides by $\map h y$, we get:
 * $\dfrac 1 {\map h y} \dfrac {\d y} {\d x} = \map g x$

Integrating both sides $x$, we get:
 * $\displaystyle \int \frac 1 {\map h y} \frac {\d y} {\d x} \rd x = \int \map g x \rd x$

which, from Integration by Substitution, reduces to the result.

The arbitrary constant $C$ happens during the integration process.

Also presented as
Some sources present this as an equation in the form:
 * $\dfrac {\d y} {\d x} = \dfrac {\map g x} {\map h y}$

whose general solution is found by solving the integration:
 * $\displaystyle \int \map h y \rd y = \int \map g x \rd x + C$

Mnemonic Device
As derivatives are not fractions, the following is a only.

This is an an abuse of notation that is likely to make some Calculus professors upset.

But it's useful.