3.15.2.6 Example \(4x^{2}y^{\prime \prime }+4xy^{\prime }+\left ( x-4\right ) y=0\)

Dividing by \(4\)

\[ x^{2}y^{\prime \prime }+xy^{\prime }+\left ( \frac {1}{4}x-1\right ) y=0 \]

Comparing the above to (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 ) & =1\\ \beta ^{2}\gamma ^{2}x^{2\gamma } & =\frac {1}{4}x\\ \left ( n^{2}\gamma ^{2}-\alpha ^{2}\right ) & =1 \end{align*}

Which implies \(\alpha =0,2\gamma =1,\beta ^{2}\gamma ^{2}=\frac {1}{4}\). Hence \(\gamma =\frac {1}{2}\) and \(\beta =1\). Last equation now says \(n^{2}\gamma ^{2}=1\) or \(n=2\). Hence 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 ) \\ & =c_{1}J_{2}\left ( \sqrt {x}\right ) +c_{2}Y_{2}\left ( \sqrt {x}\right ) \end{align*}