method of undetermined coefficients

Given a (usually non-homogeneous) ordinary differential equationMathworldPlanetmath


the method of undetermined coefficients is a way of finding an exact solution when a guess can be made as to the general form of the solution.

In this method, the form of the solution is guessed with unknown coefficients left as variables. A typical guess might be of the form Ae2x or Ax2+Bx+C. This can then be substituted into the differential equation and solved for the coefficients. Obviously the method requires knowing the approximate form of the solution, but for many problems this is a feasible requirement.

This method is most commonly used when the formula is some combination of exponentialsPlanetmathPlanetmath, polynomials, sin and cos.


Suppose we have the following second order non-homogeneous equation


If we guess that the soution is of the form f(x)=Ae2x, then, by substitution, we get


and therefore Ae2x=2e2x, so A=2, giving f(x)=2e2x as a solution.

Title method of undetermined coefficients
Canonical name MethodOfUndeterminedCoefficients
Date of creation 2013-03-22 12:52:55
Last modified on 2013-03-22 12:52:55
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 34-00
Related topic ODE
Related topic DifferentialEquation