Bessel's Equation

We recognize that this DE is Bessel's Equation with \( \nu ^2 = \cfrac{25}{4}\). Thus \( \nu = \cfrac{5}{2} \) and the solution is:

\( y= c_1 J_{\cfrac{5}{2}}(x) + c_2 J_{-\cfrac{5}{2}}(x) \)

Note that we can also use the Second Kind solution:

\( y= c_1 J_{\cfrac{5}{2}}(x) + c_2 Y_{\cfrac{5}{2}}(x) \)

A Parametric Bessel equation in the form:

\( x^2 y^{''} + xy^{'} + (\alpha ^2 x^2 - \nu ^2 )y = 0 \)

Has solution:

\( y= c_1 J_{\nu}(\alpha x) + c_2 Y_{\nu}(\alpha x) \)

A Modified Bessel equation in the form:

\( x^2 y^{''} + xy^{'} - (x^2 - \nu ^2 )y = 0 \)

Has solution:

\( y= c_1 J_{\nu}(i x) + c_2 Y_{\nu}( ix) \)

However, this is expressed in terms of complex numbers. The Modified Bessel Function of first and second kind ( \( I_{\nu}, K_{\nu} \) ) can be used instead:

\( y= c_1 I_{\nu}(x) + c_2 K_{\nu}(x) \)

Finally, another form:

\(y^{''} + \cfrac{1-2a}{x}y^{'} + (b^2c^2x^{2c-2} + \cfrac{a^2-p^2c^2}{x^2})y = 0, p \ge 0 \)

Has a convienent solution:

\( y = x^a (c_1J_{p}(bx^c) + c_2Y_{p}(bx^c)) \)

First divide by \( x^2 \) to match the form above:

\( y^{''} + \cfrac{3}{x}y^{'} + (4x^2 - \cfrac{9}{x^2})y = 0 \)

We can identify the coefficients starting with the term on y^{'} then the two on y :

\( \cfrac{1-2a}{x} = \cfrac{3}{x} \Rightarrow \bbox[5px,border:1px solid black]{a = -1}\)

\( b^2c^2x^{2c-2} = 4x^2 \Rightarrow \bbox[5px,border:1px solid black]{c = 2}, b^2 = 1 \), take \(\bbox[5px,border:1px solid black]{b=1}\)

\( \cfrac{a^2-p^2c^2}{x^2} = - \cfrac{9}{x^2} \Rightarrow p^2 = \frac{10}{4}\), take \(\bbox[5px,border:1px solid black]{p=\frac{\sqrt{10}}{2}}\)

Finally, the solution is:

\( y = x^{-1} (c_1J_{\frac{\sqrt{10}}{2}}(x^2) + c_2Y_{\frac{\sqrt{10}}{2}}(x^2)) \)