separation of variables


Separation of variablesMathworldPlanetmath is a valuable tool for solving differential equationsMathworldPlanetmath of the form

dydx=f(x)g(y)

The above equation can be rearranged algebraically through Leibniz notation, treating dy and dx as differentialsMathworldPlanetmath, to separate the variables and be conveniently integrable on both sides.

dyg(y)=f(x)dx

Instead of using differentials, we can also make use of the change of variables theorem for integration and the fact that if two integrable functions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath, then their primitives differ by a constant C. Here, we write y=y(x) and dydx=y(x) for clarity. The above equation then becomes:

y(x)g(y(x))=f(x)

Integrating both sides over x gives us the desired result:

y(x)g(y(x))𝑑x=f(x)𝑑x+C

By the change of variables theorem of integration, the left hand side is equivalent to an integral in the variable y:

dyg(y)=f(x)𝑑x+C

It follows then that

dyg(y)=F(x)+C

where F(x) is an antiderivative of f and C is the constant differencePlanetmathPlanetmath between the two primitives. 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 200 organisms increases at a rate of 15% each year. We then have a differential equation

dPdt=P+0.15P=1.15P

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

dPP=1.15𝑑t

This is just lnP=1.15t+C or

P=Ce1.15t

When we substitute P(0)=200, we see that C=200. This is where we get the general relationMathworldPlanetmath of exponential growth

P(t)=P0ekt
Title separation of variables
Canonical name SeparationOfVariables
Date of creation 2013-03-22 12:29:24
Last modified on 2013-03-22 12:29:24
Owner slider142 (78)
Last modified by slider142 (78)
Numerical id 9
Author slider142 (78)
Entry type AlgorithmMathworldPlanetmath
Classification msc 34A30
Classification msc 34A09
Classification msc 34A05
Related topic LinearDifferentialEquationOfFirstOrder
Related topic InverseLaplaceTransformOfDerivatives
Related topic SingularSolution
Related topic ODETypesReductibleToTheVariablesSeparableCase