Introduction This gives examples of converting (when possible) a second order linear ode to Bessel form. Bessel ODE is

\begin {equation} x^{2}y^{\prime \prime }+xy^{\prime }+\left ( x^{2}-n^{2}\right ) y=0 \tag {A} \end {equation} Where \(n\) is the order which can be integer or non-integer. This comes out when doing separation of variables for the Laplace and Helmholtz PDE in spherical and cylindrical coordinates. \(n\) is integer for cylindrical coordinates and half integer values (\(n=\frac {1}{2}+\mathbb {Z} \)), for spherical coordinates. \(n\) can also be any other real value. The case \(n=\frac {1}{2}+\mathbb {Z} \) is special in that the solution of the ode is reducible to standard trigonometric functions and complex exponential function. In all other cases, the solution remains in terms of Bessel functions.

The solution to (A) is known to be\[ y\left ( x\right ) =c_{1}J_{n}\left ( x\right ) +c_{2}Y_{n}\left ( x\right ) \] Where \(J_{n}\left ( x\right ) \) is Bessel function of first kind (order \(n\)). And \(Y_{n}\left ( x\right ) \) Bessel function of second kind (order \(n\)).

There is also the modified Bessel ODE which differ by a sign\begin {equation} x^{2}y^{\prime \prime }+xy^{\prime }-\left ( x^{2}+n^{2}\right ) y=0 \tag {B} \end {equation} There is however a generalized form of (A,B). Which will be used below. (Bowman 1958). This form is\begin {equation} 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 \tag {C} \end {equation} Which is obtained by applying the transformation \(\eta =\frac {y}{x^{\alpha }},\xi =\beta x^{\gamma }\) to (A). The above has the solution\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 ) \qquad \text {integer }n\tag {C1}\\ 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 ) \qquad \text {noninteger }n \tag {C2} \end {align}