3.15.2.11 Example \(x^{2}y^{\prime \prime }+xy^{\prime }+(x^{2}-5)y=0\)

\begin{equation} x^{2}y^{\prime \prime }+xy^{\prime }+(x^{2}-5)y=0 \tag {1}\end{equation}

Using the generalized form of Bessel ode

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

Which is given by (Bowman 1958)

\begin{equation} 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 \tag {C}\end{equation}

Comparing (1) and (C) shows that

\begin{align} \left ( 1-2\alpha \right ) & =1\tag {2}\\ \beta ^{2}\gamma ^{2}x^{2\gamma } & =x^{2}\tag {3}\\ \left ( n^{2}\gamma ^{2}-\alpha ^{2}\right ) & =5 \tag {4}\end{align}

(2) gives \(\alpha =0\). (3) gives \(\gamma =1\) and \(\beta ^{2}\gamma ^{2}=1\) or \(\beta =1\). Now (4) gives \(n^{2}\gamma ^{2}=5\) or \(n=\sqrt {5}\).But the solution to (C) is given 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

\[ y\left ( x\right ) =c_{1}J_{\sqrt {5}}\left ( x\right ) +c_{2}Y_{\sqrt {5}}\left ( x\right ) \]

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