3.15.2.3 Example \(x^{2}y^{\prime \prime }+bxy^{\prime }+\left ( x^{2}-v^{2}\right ) y=0\)

\begin{equation} x^{2}y^{\prime \prime }+bxy^{\prime }+\left ( x^{2}-v^{2}\right ) y=0 \tag {1}\end{equation}

Comparing (1) to the generalized form (C) \(x^{2}y^{\prime \prime }+\left ( 1-2\alpha \right ) xy^{\prime }+\left ( \beta ^{2}\gamma ^{2}x^{2\gamma }-\left ( n^{2}\gamma ^{2}-\alpha ^{2}\right ) \right ) y=0\) shows that

\begin{align*} \left ( 1-2\alpha \right ) & =b\\ 2\gamma & =2\\ \beta ^{2}\gamma ^{2} & =1\\ \left ( n^{2}\gamma ^{2}-\alpha ^{2}\right ) & =v^{2}\end{align*}

Hence \(\gamma =1,\beta =1\,.\) From first equation \(\alpha =\frac {1}{2}\left ( 1-b\right ) \). Using this in the last equation gives

\begin{align*} n^{2}-\frac {1}{4}\left ( 1-b\right ) ^{2} & =v^{2}\\ n & =\sqrt {v^{2}+\frac {1}{4}\left ( 1-b\right ) ^{2}}\end{align*}

Therefore the solution (C1) is

\begin{align*} 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 ) \\ & =x^{\frac {1}{2}\left ( 1-b\right ) }\left ( c_{1}J_{n}\left ( x\right ) +c_{2}Y_{n}\left ( x\right ) \right ) \end{align*}

For example, if \(b=4\), then the ode is \(x^{2}y^{\prime \prime }+4xy^{\prime }+\left ( x^{2}-v^{2}\right ) y=0\) and the solution is

\[ y\left ( x\right ) =x^{-\frac {3}{2}}\left ( c_{1}J_{n}\left ( x\right ) +Y_{n}\left ( x\right ) \right ) \]

Where \(n=\frac {1}{2}\sqrt {\frac {4v^{2}+9}{2}}\).