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Wednesday, 27 March 2013

Reduction of Order Technique


This technique is very important since it helps one to find a second solution independent from a known one. , 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
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