# Method of Variation of Parameters

This method is interesting whenever the previous method does not apply (when g(x) is not of the desired form). The general idea is similar to what we did for second order linear equations except that, in that case, we were dealing with a small system and here we may be dealing with a bigger one (depending on the order of the differential equation). Let us describe the general case (constant coefficients or not). Consider the equation

Suppose that a set of independent solutions of the associated homogeneous equation is known. Then a particular solution can be found as

where the functions can be obtained from the following system:

.

The determinant of this system is the Wronskian of , which is not zero. Cramer's formulas will give

,

where W(x) is the Wronskian and is the determinant obtained from the Wronskian W by replacing the -column in the vector column (0,0,..,0,1). Consequently, a particular solution to the equation (NH) is given by

Note that when the order of the equation is not high, you may want to solve the system using techniques other than Cramer's formulas.

Example: Find a particular solution of

Solution: Let us follow these steps:

(1)
Characteristic equation

Since , the roots of the characteristic equation are . Therefore, a set of independent solutions is ;

(2)
A particular solution is given by , where are solutions of the system

;

(3)
The resolution of the system gives

After integration we get

;

(4)
A particular solution is given by

Note that the constant 1 in may be dropped since it is the solution of the associated homogeneous equation.

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