PlanetMath: separation of variables

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separation of variables

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Separation of variables is a valuable tool for solving differential equations of the form

$\displaystyle \frac{dy}{dx}=f(x)g(y)$

The above equation can be rearranged algebraically through Leibniz notation to separate the variables and be conveniently integrable.

$\displaystyle \frac{dy}{g(y)}=f(x)dx$

It follows then that

$\displaystyle \int\frac{dy}{g(y)} = F(x) + C$

where

and

is a constant of integration. This gives a general form of the solution. An explicit form may be derived by an initial value.

Example: A population that is initially at organisms increases at a rate of $15\%$ each year. We then have a differential equation

$\displaystyle \frac{dP}{dt} = 0.15P$

The solution of this equation is relatively straightforward, we simple separate the variables algebraically and integrate.

$\displaystyle \int \frac{dP}{P} = \int 0.15\;dt$

This is just $\ln P = 0.15t + C$ or

$\displaystyle P=Ce^{0.15t}$

When we substitute

, we see that

. This is where we get the general relation of exponential growth

$\displaystyle P(t) = P_0e^{kt}$

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Cross-references: exponential growth, relation, simple, solution, constant of integration, antiderivative, variables, Leibniz notation, equation, differential equations
There are 10 references to this entry.

This is version 2 of separation of variables, born on 2002-02-26, modified 2002-02-26.
Object id is 2712, canonical name is SeparationOfVariables.
Accessed 9753 times total.

Classification:

AMS MSC:	34A05 (Ordinary differential equations :: General theory :: Explicit solutions and reductions)
	34A09 (Ordinary differential equations :: General theory :: Implicit equations, differential-algebraic equations)
	34A30 (Ordinary differential equations :: General theory :: Linear equations and systems, general)

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