Separation of Variables

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

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

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

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

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

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

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

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.