If n is an integer, use the substitution to show that the

Chapter 6, Problem 54E

(choose chapter or problem)

If n is an integer, use the substitution \(R(x)=(\alpha x)^{-1 / 2} Z(x)\) to show that the general solution of the differential equation

\(x^{2} R^{\prime \prime}+2 x R^{\prime}+\left[\alpha^{2} x^{2}-n(n+1)\right] R=0\)

on the interval \((0, \infty)\) is \(R(x)=c_{1} j_{n}(\alpha x)+c_{2} y_{n}(\alpha x)\), where \(j_{n}(\alpha x)\) and \(y_{n}(\alpha x)\) are the spherical Bessel functions of the first and second kind defined in (27).

Text Transcription:

R(x)=(alphax)^-1/2Z(x)

x^2R^prime\prime+2xR^prime+[alpha^2x^2-n(n+1)] R=0

(0,infty)

R(x)=c_1j_n(alphax)+c_2y_n(alphax)

j_n(alphax)

y_n(alphax)