Reduction of Order Technique

This technique is very important since it helps one to find a second solution independent from a known one. Therefore, according to the previous section, in order to find the general solution to y'' + p(x)y' + q(x)y = 0, we need only to find one (non-zero) solution, tex2html_wrap_inline143 .

Let tex2html_wrap_inline143 be a non-zero solution of

displaymath147

Then, a second solution tex2html_wrap_inline149 independent of tex2html_wrap_inline143 can be found as

displaymath135

Easy calculations give

displaymath136,

where C is an arbitrary non-zero constant. Since we are looking for a second solution one may take C=1, to get

displaymath157

Remember that this formula saves time. But, if you forget it you will have to plug tex2html_wrap_inline159 into the equation to determine v(x) which may lead to mistakes !
The general solution is then given by

displaymath137

Example: Find the general solution to the Legendre equation

displaymath163,

using the fact that tex2html_wrap_inline165 is a solution.

Solution: It is easy to check that indeed tex2html_wrap_inline165 is a solution. First, we need to rewrite the equation in the explicit form

displaymath169

We may try to find a second solution tex2html_wrap_inline171 by plugging it into the equation. We leave it to the reader to do that! Instead let us use the formula

displaymath173

Techniques of integration (of rational functions) give

displaymath175,

which gives

displaymath177

The general solution is then given by

displaymath179

Remark: The formula giving tex2html_wrap_inline149 can be obtained by also using the properties of the Wronskian (see also the discussion on the Wronskian).

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

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