# 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:

$\displaystyle \int \frac {\d y} {\map h y} = \int \map g x \rd x + C$

### General Result

Suppose a first order ordinary differential equation can be expressible in this form:

$g_1 \left({x}\right) h_1 \left({y}\right) + g_2 \left({x}\right) h_2 \left({y}\right) \dfrac {\d y} {\d x} = 0$

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

Its general solution is found by solving the integration:

$\displaystyle \int \frac {g_1 \left({x}\right)} {g_2 \left({x}\right)} \rd x + \int \frac {h_2 \left({y}\right)} {h_1 \left({y}\right)} \rd y = C$

## 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 with respect to $x$, 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.

$\blacksquare$

## 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 mnemonic device only.

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

But it's useful.

 $\displaystyle \frac {\d y} {\d x}$ $=$ $\displaystyle \map g x \, \map h y$ $\displaystyle \leadsto \ \$ $\displaystyle \d y$ $=$ $\displaystyle \map g x \, \map h y \rd x$ solve for $\d y$ $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 {\map h y} \rd y$ $=$ $\displaystyle \map g x \rd x$ collecting like terms on each side

## Historical Note

The method of Separation of Variables was described by Johann Bernoulli between the years $1694$ – $1697$.