3.15.2.10 Example \(f^{\prime \prime }+\frac {\lambda }{x}f^{\prime }-\mu f=0\)

Multiplying by \(x^{2}\)

\begin{equation} x^{2}f^{\prime \prime }+\lambda xf^{\prime }+\left ( -\mu x^{2}\right ) f=0 \tag {1}\end{equation}

Using the generalized form of Bessel ode

\begin{equation} x^{2}f^{\prime \prime }+xf^{\prime }+\left ( x^{2}-n^{2}\right ) f=0 \tag {A}\end{equation}

Which is given by (Bowman 1958)

\begin{equation} x^{2}f^{\prime \prime }+\left ( 1-2\alpha \right ) xf^{\prime }+\left ( \beta ^{2}\gamma ^{2}x^{2\gamma }-\left ( n^{2}\gamma ^{2}-\alpha ^{2}\right ) \right ) f=0 \tag {C}\end{equation}

Comparing (1) and (C) shows that

\begin{align} \left ( 1-2\alpha \right ) & =\lambda \tag {2}\\ \beta ^{2}\gamma ^{2}x^{2\gamma } & =-\mu x^{2}\tag {3}\\ \left ( n^{2}\gamma ^{2}-\alpha ^{2}\right ) & =0 \tag {4}\end{align}

(2) gives \(\alpha =\frac {1}{2}-\frac {1}{2}\lambda \). (3) gives \(2\gamma =2\) or \(\gamma =1\). And (3) also shows that \(\beta ^{2}\gamma ^{2}=-\mu \) or \(\beta =\sqrt {-\mu }\). Now (4) gives \(\left ( n^{2}-\left ( \frac {1}{2}-\frac {1}{2}\lambda \right ) ^{2}\right ) =0\) or \(n=\left ( \frac {1}{2}-\frac {1}{2}\lambda \right ) \). (taking positive root). But the solution to (C) is gives by

\[ y\left ( x\right ) =x^{\alpha }\left ( c_{1}J_{n}\left ( \beta x^{\gamma }\right ) +c_{2}Y_{n}\left ( \beta x^{\gamma }\right ) \right ) \]

Therefore the solution to (1) is

\[ y\left ( x\right ) =x^{\left ( \frac {1}{2}-\frac {1}{2}\lambda \right ) }\left ( c_{1}J_{\left ( \frac {1}{2}-\frac {1}{2}\lambda \right ) }\left ( \sqrt {-\mu }x\right ) +c_{2}Y_{\left ( \frac {1}{2}-\frac {1}{2}\lambda \right ) }\left ( \sqrt {-\mu }x\right ) \right ) \]

Where \(J\) is the Bessel function of first kind and \(Y\) is the Bessel function of the second kind.