Chapter 12.2, Problem 4P

Use the method of reduction of order [Chapter 8, Section 7 (e)] and the known solution $P_i(x)$ of Legendre’s equation to find the second solution $Q_l(x)$ (in terms of an integral). Evaluate the integral for the cases $l=0$ and $l=1$ to $\ln Q_0$ and $Q_1$ Note the divergence of the logarithms at $x=\pm 1.$ Expand the logarithms in $Q_0$ to get the divergent series mentioned above .