3.15.2.2 Example \(x^{2}y^{\prime \prime }+xy^{\prime }+xy=0\)

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

Comparing (1) to (C) shows that

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

Hence

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

Which implies

\begin{align} 2\gamma & =1\tag {4}\\ \beta ^{2}\gamma ^{2} & =1 \tag {5}\end{align}

(2) gives \(\alpha =0\). (4) gives \(\gamma =\frac {1}{2}\). Substituting these into (3) gives

\[ n=0 \]

And (5) gives \(\beta ^{2}=4\) or \(\beta =\pm 2\). Therefore from (C1) the solution 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 ) \\ & =c_{1}J_{0}\left ( 2\sqrt {x}\right ) +c_{2}Y_{n}\left ( 2\sqrt {x}\right ) \end{align*}