Method of Variation of Parameters
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:
a set of fundamental solutions of the
associated homogeneous equation
; ; and ;
and into the equation giving the particular
solution.
Example: Find the particular solution to
Solution: Let us follow the steps:
;
;
and 
, and get
Using techniques of integration, we get
;
,
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.
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