3.15.2.9 Example \(y^{\prime \prime }-x^{-\frac {1}{4}}y=0\)

Multiplying by \(x^{\frac {1}{4}}\)

\[ x^{\frac {1}{4}}y^{\prime \prime }-y=0 \]

Multiplying by \(x^{\frac {7}{4}}\)

\[ x^{2}y^{\prime \prime }-x^{\frac {7}{4}}y=0 \]

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 ) & =0\\ \beta ^{2}\gamma ^{2}x^{2\gamma } & =-x^{\frac {7}{4}}\\ \left ( n^{2}\gamma ^{2}-\alpha ^{2}\right ) & =0 \end{align*}

Which implies \(\alpha =\frac {1}{2}\) and \(2\gamma =\frac {7}{4}\) or \(\gamma =\frac {7}{8}\) and \(\beta ^{2}\gamma ^{2}=-1\) or \(\beta ^{2}=-\frac {1}{\left ( \frac {7}{8}\right ) ^{2}}=-\frac {64}{49}\). Hence \(\beta =i\frac {8}{7}\). Last equation now becomes \(\left ( n^{2}\left ( \frac {49}{64}\right ) -\frac {1}{4}\right ) =0\), or \(n=\frac {4}{7}\). Hence the solution (C2) for non integer \(n\) becomes

\begin{align*} y\left ( x\right ) & =x^{\alpha }\left ( c_{1}J_{n}\left ( \beta x^{\gamma }\right ) +c_{2}J_{-n}\left ( \beta x^{\gamma }\right ) \right ) \\ & =\sqrt {x}\left ( c_{1}J_{\frac {4}{7}}\left ( i\frac {8}{7}x^{\frac {7}{8}}\right ) +c_{2}J_{-\frac {4}{7}}\left ( i\frac {8}{7}x^{\frac {7}{8}}\right ) \right ) \end{align*}