separation of variables

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


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.


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:


Integrating both sides over x gives us the desired result:


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


It follows then that


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


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


This is just lnP=1.15t+C or


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

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