# Method of Variation of Parameters This method has no prior conditions to be satisfied. Therefore, it may sound more general than the previous method. We will see that this method depends on integration while the previous one is purely algebraic which, for some at least, is an advantage.

Consider the equation In order to use the method of variation of parameters we need to know that is a set of fundamental solutions of the associated homogeneous equation y'' + p(x)y' + q(x)y = 0. We know that, in this case, the general solution of the associated homogeneous equation is . The idea behind the method of variation of parameters is to look for a particular solution such as where and are functions. From this, the method got its name.
The functions and are solutions to the system ,

which implies ,

where is the wronskian of and . Therefore, we have Summary:Let us summarize the steps to follow in applying this method:

(1)
Find a set of fundamental solutions of the associated homogeneous equation

y'' + p(x)y' + q(x)y = 0.
;

(2)
Write down the form of the particular solution ;

(3)
Write down the system ;

(4)
Solve it. That is, find and ;
(5)
Plug and into the equation giving the particular solution.

Example: Find the particular solution to Solution: Let us follow the steps:

(1)
A set of fundamental solutions of the equation y'' + y = 0 is ;
(2)
The particular solution is given as (3)
We have the system ;

(4)
We solve for and , and get Using techniques of integration, we get ;

(5)
The particular solution is: ,

or Remark: Note that since the equation is linear, we may still split if necessary. For example, we may split the equation ,

into the two equations then, find the particular solutions for (1) and for (2), to generate a particular solution for the original equation by There are no restrictions on the method to be used to find or . For example, we can use the method of undetermined coefficients to find , while for , we are only left with the variation of parameters. [Differential Equations] [First Order D.E.] [Second Order D.E.]
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