# Separation of Variables

## Contents

## 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 $1694$ – $1697$.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 2.7$: Homogeneous Equations - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**differential equation**

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