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
where is the wronskian of and . Therefore, we have
Summary:Let us summarize the steps to follow in applying this method:
Example: Find the particular solution to
Solution: Let us follow the steps:
Using techniques of integration, we get
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.
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Author: Mohamed Amine Khamsi