The following DE is called Bessel's Equation:
\( x^2 y^{''} + xy^{'} + (x^2 - \nu ^2 )y = 0 \)
Note that is is a second-order linear equation with non-constant coefficients. The solution to this DE can be expressed in terms of the First and Second Kind of Bessel Functions.
When \(\nu\) is not an integer, the solution is:
\(y = c_1 J_{\nu}(x) + c_2 J_{-\nu}(x) \)
When \(\nu\) is an integer (also works if it is not an integer), the solution is:
\(y= c_1 J_{\nu}(x) + c_2 Y_{\nu}(x) \)
What is the solution to the DE \( x^2 y^{''} + xy^{'} + (x^2 - \cfrac{25}{4} )y = 0 \) ?
We recognize that this DE is Bessel's Equation with \(\nu ^2 = \cfrac{25}{4}\). Thus \(\nu = \cfrac{5}{2} \) and the solution is:
\(y= c_1 J_{\frac{5}{2}}(x) + c_2 J_{-\frac{5}{2}}(x) \)
Note that we can also use the Second Kind solution:
\(y= c_1 J_{\frac{5}{2}}(x) + c_2 Y_{\frac{5}{2}}(x)\)
A Parametric Bessel equation in the form:
\( x^2 y^{''} + xy^{'} + (\alpha ^2 x^2 - \nu ^2 )y = 0 \)
Has the following 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 the following 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:
\(y = x^a (c_1J_{p}(bx^c) + c_2Y_{p}(bx^c))\)
Determine the solution to the DE: \( x^2y^{''} + 3xy^{'} + (4x^4 - 9)y = 0\).
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 \(\bbox[5px,border:1px solid black]{b=1}\)
\( \cfrac{a^2-p^2c^2}{x^2} = - \cfrac{9}{x^2} \Rightarrow p^2 = \cfrac{10}{4}\), take \(\bbox[5px,border:1px solid black]{p=\cfrac{\sqrt{10}}{2}}\)
Finally, the solution is:
\(y = x^{-1} (c_1J_{\frac{\sqrt{10}}{2}}(x^2) + c_2Y_{\frac{\sqrt{10}}{2}}(x^2))\)