(Legendre functions of second kind)For the Legendre equation (7) on the interval −1 < x...

(Legendre functions of second kind)For the Legendre equation (7) on the interval −1 < x < 1, we obtained the bounded solutiony(x) =Pn(x).In this exercise we seek a second LI solution, denoted asQn(x) and called theLegendre function of the second kind.Then the general solution of (7) can be expressed as

y (x)= APn(x)+ BQn(x).(11.1)

More generally, consider any nonnegative integern. With onlyPn(x)in hand, seek a second solution (by reduction of order) in the formy(x)=A(x)Pn(x), and show thatQn(x)is given by