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