# 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).

[Differential Equations] [First Order D.E.] [Second Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi