variation of parameters

The method of variation of parametersMathworldPlanetmath is a way of finding a particular solution to a nonhomogeneous linear differential equation.

Suppose that we have an nth order linear differential operatorMathworldPlanetmath


and a corresponding nonhomogeneous differential equation

L[y]=g(t). (1)

Suppose that we know a fundamental set of solutions y1,y2,,yn of the corresponding homogeneous differential equation L[yc]=0. The general solution of the homogeneous equation is


where c1,c2,,cn are constants. The general solution to the nonhomogeneous equation L[y]=g(t) is then


where Y(t) is a particular solution which satisfies L[Y]=g(t), and the constants c1,c2,,cn are chosen to satisfy the appropriate boundary conditionsMathworldPlanetmath or initial conditions.

The key step in using variation of parameters is to suppose that the particular solution is given by

Y(t)=u1(t)y1(t)+u2(t)y2(t)++un(t)yn(t), (2)

where u1(t),u2(t),,un(t) are as yet to be determined functions (hence the name variation of parameters). To find these n functions we need a set of n independent equations. One obvious condition is that the proposed ansatz satisfies Eq. (1). Many possible additional conditions are possible, we choose the ones that make further calculations easier. Consider the following set of n-1 conditions

y1u1+y2u2++ynun = 0
y1u1+y2u2++ynun = 0
y1(n-2)u1+y2(n-2)u2++yn(n-2)un = 0.

Now, substituting Eq. (2) into L[Y]=g(t) and using the above conditions, we can get another equation


So we have a system of n equations for u1,u2,,un which we can solve using Cramer’s rule:


Such a solution always exists since the Wronskian W=W(y1,y2,,yn) of the system is nowhere zero, because the y1,y2,,yn form a fundamental set of solutions. Lastly the term Wm is the Wronskian determinant with the mth column replaced by the column (0,0,,0,1).

Finally the particular solution can be written explicitly as



Title variation of parameters
Canonical name VariationOfParameters
Date of creation 2013-03-22 12:39:16
Last modified on 2013-03-22 12:39:16
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type TheoremMathworldPlanetmath
Classification msc 34A30
Classification msc 34A05
Synonym variation of constants