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
This section needs to be rewritten as a separate proof from the standpoint of differentials.
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
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
- For a video presentation of the contents of this page, visit the Khan Academy.
