# Checking Qn(x) is a solution to legendre eq. (n=3)

1. Apr 19, 2012

### linda300

Hey,

I have a question which ends by asking to verify that Q3(x) is a solution to the legendre equation,

I took the first and second derivatives of it and before I continue with this messy verification I wanted to know if there was a simpler way to check.

Q3(x) = (1/4)x(5x^2 - 3)log((1+x)/(1-x)),

Is there some quick way to verify its a solution to the n=3 legendre equation? Rather then substituting it straight into the equation?

2. Apr 21, 2012

### vela

Staff Emeritus
Your expression for Q3(x) seems to be missing a few terms.

Note that $P_3(x) = \frac{1}{2}(5x^3-3x)$ is a factor in the last term. Instead of multiplying everything out and grinding it out, use the fact that P3(x) satisfies the differential equation.