Separation of Variables

Theorem
Suppose a first order ordinary differential equation can be expressible in this form:
 * $\dfrac {\mathrm d y} {\mathrm d x} = g \left({x}\right) h \left({y}\right)$

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

Its general solution is found by solving the integration:
 * $\displaystyle \int \frac {\mathrm d y} {h \left({y}\right)} = \int g \left({x}\right) \, \mathrm d x + C$

Proof
Dividing both sides by $h \left({y}\right)$, we get:
 * $\dfrac 1 {h \left({y}\right)} \dfrac {\mathrm d y} {\mathrm d x} = g \left({x}\right)$

Integrating both sides $x$, we get:
 * $\displaystyle \int \frac 1 {h \left({y}\right)} \frac {\mathrm d y} {\mathrm d x} \, \mathrm d x = \int g \left({x}\right) \, \mathrm d x$

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:
 * $\dfrac {\mathrm d y} {\mathrm d x} = \dfrac {g \left({x}\right)} {h \left({y}\right)}$

whose general solution is found by solving the integration:
 * $\displaystyle \int h \left({y}\right) \, \mathrm d y = \int g \left({x}\right) \, \mathrm d x + C$

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

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

But it's useful.