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.

**
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