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

\begin{equation} y^{\prime \prime }-\frac {1}{x}y=0 \tag {1}\end{equation}

Multiplying both sides by \(x^{2}\) gives

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

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

First equation gives \(\alpha =\frac {1}{2}\). Second equation gives \(\gamma =\frac {1}{2}\) and \(\beta ^{2}\gamma ^{2}=-1\). Therefore \(\beta ^{2}=-4\) or \(\beta =\pm 2i\). Last equation gives \(n^{2}\gamma ^{2}=\frac {1}{4}\) or \(n=1\) since \(\gamma ^{2}=\frac {1}{4}\). 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_{1}\left ( 2i\sqrt {x}\right ) +c_{2}Y_{1}\left ( 2i\sqrt {x}\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

\[ y\left ( x\right ) =\sqrt {x}\left ( ic_{1}I_{1}\left ( 2\sqrt {x}\right ) +c_{2}Y_{1}\left ( 2i\sqrt {x}\right ) \right ) \]

Alternative longer method is the following:

Trying standard transformation  \(y=\sqrt {x}Y\). The ode becomes

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

Using the transformation \(x=t^{2}\) the above becomes

\[ t^{2}Y^{\prime \prime }+tY^{\prime }-\left ( 4t^{2}+1\right ) Y=0 \]

Finally applying the standard transformation \(t=\frac {1}{2}z\) to fix the term \(\left ( 4t^{2}+1\right ) \) to standard form the above becomes

\[ z^{2}Y^{\prime \prime }+zY^{\prime }-\left ( t^{2}+1\right ) Y=0 \]

This is modified Bessel ODE whose solution is known to be

\[ Y\left ( z\right ) =c_{1}I_{1}\left ( z\right ) +c_{2}K_{1}\left ( z\right ) \]

Where \(I_{1}\) is modified Bessel function of first kind and \(K_{1}\) is modified Bessel function of second kind. But \(z=2t\). Hence the above becomes

\[ Y\left ( t\right ) =c_{1}I_{1}\left ( 2t\right ) +c_{2}K_{1}\left ( 2t\right ) \]

But \(t=\sqrt {x}\). The above becomes

\[ Y\left ( x\right ) =c_{1}I_{1}\left ( 2\sqrt {x}\right ) +c_{2}K_{1}\left ( 2\sqrt {x}\right ) \]

But \(y\left ( x\right ) =\sqrt {x}Y\left ( z\right ) \) hence

\[ y\left ( x\right ) =c_{1}\sqrt {x}I_{1}\left ( 2\sqrt {x}\right ) +c_{2}\sqrt {x}K_{1}\left ( 2\sqrt {x}\right ) \]