Chapter 12.23, Problem 25MP

In Problem 22.26, replace $x$ by $x/n$ in the $y$ differential equation and set $\lambda =n$ to show that the differential equation satisfied by the functions $f_n(x)$ in Problem 22.27 is $y^{\prime \prime }+\left(\frac{1}{x}-\frac{1}{4n^2}-\frac{l(l+1)}{x^2}\right)y=0$.

Hence show by Problem 24 that the functions $f_n(x)$ are orthogonal on $(0,\infty )$.