Method of Variation of Parameters

This method is interesting whenever the previous method does not apply (when g(x) is not of the desired form). The general idea is similar to what we did for second order linear equations except that, in that case, we were dealing with a small system and here we may be dealing with a bigger one (depending on the order of the differential equation). Let us describe the general case (constant coefficients or not). Consider the equation

displaymath407

Suppose that a set of independent solutions tex2html_wrap_inline419 of the associated homogeneous equation is known. Then a particular solution can be found as

displaymath408

where the functions tex2html_wrap_inline421 can be obtained from the following system:

displaymath409.

The determinant of this system is the Wronskian of tex2html_wrap_inline423, which is not zero. Cramer's formulas will give

displaymath410,

where W(x) is the Wronskian tex2html_wrap_inline427 and tex2html_wrap_inline429 is the determinant obtained from the Wronskian W by replacing the tex2html_wrap_inline433 -column in the vector column (0,0,..,0,1). Consequently, a particular solution to the equation (NH) is given by

displaymath411

Note that when the order of the equation is not high, you may want to solve the system using techniques other than Cramer's formulas.

Example: Find a particular solution of

displaymath439

Solution: Let us follow these steps:

(1)
Characteristic equation

displaymath441

Since tex2html_wrap_inline443 , the roots of the characteristic equation are tex2html_wrap_inline445 . Therefore, a set of independent solutions is tex2html_wrap_inline447;

(2)
A particular solution is given by tex2html_wrap_inline449, where tex2html_wrap_inline451 are solutions of the system

displaymath453;

(3)
The resolution of the system gives

displaymath455

After integration we get

displaymath457;

(4)
A particular solution is given by

displaymath459

Note that the constant 1 in tex2html_wrap_inline345 may be dropped since it is the solution of the associated homogeneous equation.

[Differential Equations] [First Order D.E.] [Second Order D.E.]
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Author: Mohamed Amine Khamsi

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