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

displaymath46

In order to use the method of variation of parameters we need to know that tex2html_wrap_inline62 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 tex2html_wrap_inline66 . The idea behind the method of variation of parameters is to look for a particular solution such as

displaymath47

where tex2html_wrap_inline68 and tex2html_wrap_inline70 are functions. From this, the method got its name.
The functions tex2html_wrap_inline68 and tex2html_wrap_inline70 are solutions to the system

displaymath48,

which implies

displaymath49,

where tex2html_wrap_inline45 is the wronskian of tex2html_wrap_inline41 and tex2html_wrap_inline43. Therefore, we have

displaymath50

Summary:Let us summarize the steps to follow in applying this method:

Example: Find the particular solution to

displaymath88

Solution: Let us follow the steps:

(1)
A set of fundamental solutions of the equation y'' + y = 0 is tex2html_wrap_inline92 ;
(2)
The particular solution is given as

displaymath94

(3)
We have the system

displaymath53 ;

(4)
We solve for tex2html_wrap_inline96 and tex2html_wrap_inline98 , and get

displaymath100

Using techniques of integration, we get

displaymath102 ;

(5)
The particular solution is:

displaymath104,

or

displaymath106

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

displaymath108,

into the two equations

displaymath110

then, find the particular solutions tex2html_wrap_inline112 for (1) and tex2html_wrap_inline116 for (2), to generate a particular solution for the original equation by

displaymath120

There are no restrictions on the method to be used to find tex2html_wrap_inline112 or tex2html_wrap_inline116 . For example, we can use the method of undetermined coefficients to find tex2html_wrap_inline112, while for tex2html_wrap_inline116, we are only left with the variation of parameters.

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Author: Mohamed Amine Khamsi

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