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,  .
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