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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