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

[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?

 

[vela]

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.