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:

$\map {g_1} x \map {h_1} y + \map {g_2} x \map {h_2} y \dfrac {\d y} {\d x} = 0$

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

Its general solution is found by solving the integration:

$\displaystyle \int \frac {\map {g_1} x} {\map {g_2} x} \rd x + \int \frac {\map {h_2} y} {\map {h_1} y} \rd y = C$

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 $C$ appears 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

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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.

$\ds \frac {\d y} {\d x}$

$=$

$\ds \map g x \, \map h y$

$\ds \leadsto \ \ $

$\ds \d y$

$=$

$\ds \map g x \, \map h y \rd x$

solve for $\d y$

$\ds \leadsto \ \ $

$\ds \frac 1 {\map h y} \rd y$

$=$

$\ds \map g x \rd x$

collecting like terms on each side

Historical Note

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

Sources

1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.7$: Homogeneous Equations
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: differential equation
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: differential equation
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: separable first-order differential equation

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