Separation of Variables

From ProofWiki
Jump to navigation Jump to search

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