3.15.2.7 Example \(y^{\prime \prime }-\frac {1}{x^{\frac {3}{2}}}y=0\)

Multiplying by \(x^{\frac {3}{2}}\)

\[ x^{\frac {3}{2}}y^{\prime \prime }-y=0 \]

Multiplying by \(x^{\frac {1}{2}}\)

\[ x^{2}y^{\prime \prime }-x^{\frac {1}{2}}y=0 \]

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 {1}{2}}\\ \left ( n^{2}\gamma ^{2}-\alpha ^{2}\right ) & =0 \end{align*}

Which implies \(\alpha =\frac {1}{2},2\gamma =\frac {1}{2},\beta ^{2}\gamma ^{2}=-1\). Hence \(\gamma =\frac {1}{4}\) and \(\beta ^{2}=-16\) or \(\beta =\pm 4i\). Last equation now says \(\left ( n^{2}\frac {1}{16}-\frac {1}{4}\right ) =0\) 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 ) \\ & =\sqrt {x}\left ( c_{1}J_{2}\left ( 4ix^{\frac {1}{4}}\right ) +c_{2}Y_{2}\left ( 4ix^{\frac {1}{4}}\right ) \right ) \end{align*}

By properties of Bessel functions, where \(J_{n}\left ( ai\sqrt {x}\right ) =i^{n}I_{n}\left ( a\sqrt {x}\right ) \), then the above becomes

\[ y\left ( x\right ) =\sqrt {x}\left ( -c_{1}I_{2}\left ( 4x^{\frac {1}{4}}\right ) +c_{2}Y_{2}\left ( 4ix^{\frac {1}{4}}\right ) \right ) \]