Reduction of Order Technique
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, 
.
Let be a non-zero solution of
Then, a second solution 
independent of can be found as
Easy calculations give
,
where C is an arbitrary non-zero constant. Since we are looking for a second solution one may take C=1, to get
Remember that this formula saves time. But, if you forget it you
will have to plug 
into the equation to determine v(x)
which may lead to mistakes !
The general solution is then given by
Example: Find the general solution to the Legendre equation
,
using the fact that 
is a solution.
Solution: It is easy to check that indeed
is a solution. First, we need to rewrite the equation in the
explicit form
We may try to find a second solution 
by plugging it into the equation. We leave it to the reader to
do that! Instead let us use the formula
Techniques of integration (of rational functions) give
,
which gives
The general solution is then given by
Remark: The formula giving can be obtained by also
using the
properties of the Wronskian (see also the discussion on the Wronskian).
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