3.9.8.5.4 Example 10. Bessel ODE

Given the ode

\begin{equation} y^{\prime \prime }\left ( x\right ) +\left ( 1-\frac {3}{4x^{2}}\right ) y\left ( x\right ) =0 \tag {A}\end{equation}

Trying change of variables on the dependent variable (first method).  In this method we assume

\[ y=v\left ( x\right ) z\left ( x\right ) \]

The ode is \(y^{\prime \prime }+py^{\prime }+qy=0\). Hence \(p=0,q=\left ( 1-\frac {3}{4x^{2}}\right ) \).  Therefore the Liouville ode invariant is

\begin{align*} q_{1} & =q-\frac {1}{2}p^{\prime }-\frac {1}{4}p^{2}\\ & =\left ( 1-\frac {3}{4x^{2}}\right ) \end{align*}

Since \(q_{1}\) is not constant, then this method does not work.

Trying change of variable on independent variable.

Let \(z=g\left ( x\right ) \) where \(z\) will be the new independent variable. In general, given an ode of the form

\[ y^{\prime \prime }\left ( x\right ) +p\left ( x\right ) y^{\prime }\left ( x\right ) +q\left ( x\right ) y\left ( x\right ) =r\left ( x\right ) \]

Then applying this transformation results in

\begin{equation} y^{\prime \prime }\left ( z\right ) +p_{1}\left ( z\right ) y^{\prime }\left ( z\right ) +q_{1}\left ( z\right ) y\left ( z\right ) =r_{1}\left ( z\right ) \tag {1}\end{equation}

Where

\begin{align} p_{1}\left ( z\right ) & =\frac {z^{\prime \prime }\left ( x\right ) +pz^{\prime }\left ( x\right ) }{\left ( z^{\prime }\left ( x\right ) \right ) ^{2}}\tag {2}\\ q_{1}\left ( z\right ) & =\frac {q}{\left ( z^{\prime }\left ( x\right ) \right ) ^{2}}\tag {3}\\ r_{1}\left ( z\right ) & =\frac {r}{\left ( z^{\prime }\left ( x\right ) \right ) ^{2}} \tag {4}\end{align}

Approach 1. Let \(q_{1}=c^{2}\) where \(c^{2}\) is some constant. This implies

\begin{align} \frac {q}{\left ( z^{\prime }\left ( x\right ) \right ) ^{2}} & =c^{2}\nonumber \\ z & =\frac {1}{c}\int \sqrt {q}dx \tag {5}\end{align}

If with this \(z\), then \(p_{1}\) turns out to be constant, then it means (1) is second order constant coefficient ode. Applying this on current ode then (5) becomes

\begin{align*} z & =\frac {1}{c}\int \sqrt {\left ( 1-\frac {3}{4x^{2}}\right ) }dx\\ & =\frac {1}{2c}\left ( \sqrt {4x^{2}-3}+\sqrt {3}\arctan \left ( \frac {\sqrt {3}}{\sqrt {4x^{2}-3}}\right ) \right ) \end{align*}

Hence (2) becomes

\begin{align*} p_{1}\left ( z\right ) & =\frac {z^{\prime \prime }\left ( x\right ) +pz^{\prime }\left ( x\right ) }{\left ( z^{\prime }\left ( x\right ) \right ) ^{2}}\\ & =\frac {6c}{\left ( 4x^{2}-3\right ) ^{\frac {3}{2}}}\end{align*}

Which is not a constant. So this transformation did not work.  So change of variable on both the dependent and independent variable does not work for this ode to convert it to one with constant coefficient. Trying converting it to standard Bessel ODE.  Using this change of variable on the dependent variable

\[ y=ux^{\frac {1}{2}}\]

To transform (A) to standard Bessel ODE

\[ x^{2}u^{\prime \prime }+xu^{\prime }+\left ( x^{2}-1\right ) u=0 \]

Since \(y=ux^{\frac {1}{2}}\) then

\begin{equation} \frac {dy}{dx}=\frac {du}{dx}x^{\frac {1}{2}}+u\frac {x^{\frac {-1}{2}}}{2} \tag {2A}\end{equation}

And

\begin{align} \frac {d^{2}y}{dx^{2}} & =\frac {d}{dx}\left ( \frac {du}{dx}x^{\frac {1}{2}}+u\frac {x^{\frac {-1}{2}}}{2}\right ) \nonumber \\ & =\frac {d}{dx}\left ( \frac {du}{dx}x^{\frac {1}{2}}\right ) +\frac {d}{dx}\left ( u\frac {x^{\frac {-1}{2}}}{2}\right ) \nonumber \\ & =\frac {d^{2}u}{dx^{2}}x^{\frac {1}{2}}+\frac {1}{2}\frac {du}{dx}x^{-\frac {1}{2}}+\frac {1}{2}\frac {du}{dx}x^{\frac {-1}{2}}-\frac {1}{4}ux^{-\frac {3}{2}}\nonumber \\ & =\frac {d^{2}u}{dx^{2}}x^{\frac {1}{2}}+\frac {du}{dx}x^{-\frac {1}{2}}-\frac {1}{4}ux^{-\frac {3}{2}} \tag {3A}\end{align}

Substituting (2A,3A) into (A) gives

\begin{align*} \frac {d^{2}u}{dx^{2}}x^{\frac {1}{2}}+\frac {du}{dx}x^{-\frac {1}{2}}-\frac {1}{4}ux^{-\frac {3}{2}}+\left ( 1-\frac {3}{4x^{2}}\right ) ux^{\frac {1}{2}} & =0\\ \frac {d^{2}u}{dx^{2}}x^{\frac {1}{2}}+\frac {du}{dx}x^{-\frac {1}{2}}-\frac {1}{4}ux^{-\frac {3}{2}}+ux^{\frac {1}{2}}-\frac {3}{4}ux^{-\frac {3}{2}} & =0\\ \frac {d^{2}u}{dx^{2}}x^{\frac {1}{2}}+\frac {du}{dx}x^{-\frac {1}{2}}-ux^{-\frac {3}{2}}+ux^{\frac {1}{2}} & =0 \end{align*}

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

\begin{align*} x^{2}\frac {d^{2}u}{dx^{2}}+x\frac {du}{dx}-u+ux^{2} & =0\\ x^{2}\frac {d^{2}u}{dx^{2}}+x\frac {du}{dx}-\left ( 1-x^{2}\right ) u & =0\\ x^{2}\frac {d^{2}u}{dx^{2}}+x\frac {du}{dx}+\left ( x^{2}-1\right ) u & =0 \end{align*}

Which is Bessel ode where order is \(n=1\). This has known standard solution. Once \(u\left ( x\right ) \) is known, then \(y\left ( x\right ) \) which is the solution to the original ODE (A) is now known also. There is a more general method and better method to find if second order ode can be transformed to Bessel ODE. See my main page for examples and description.