A more general Bessel equation?

[ZumaBird]

In the solution to a recent problem set, my prof referenced a "general Bessel ODE" which he gave in the form:

[tex]x^{2}\frac{d^{2}y}{dx^{2}}+x\left(a+2bx^{q}\right)\frac{dy}{dx}+\left[c+dx^{2s}-b\left(1-a-q\right)x^{q}+b^{2}x^{2q}\right]y=0[/tex]

The only format of the Bessel ODE that appears in the coursenotes is:

[tex]x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+\left(x^{2}-b^{2}\right)y=0[/tex]

Does anybody recognize the first form and know the general solution in terms of the modified and unmodified Bessel functions of the first and second kinds?

 

[mighty2000]

Hello,

Yes, I am familiar with the general Bessel ODE that your professor referenced. This form is known as the modified Bessel equation and can be written as:

x^{2}\frac{d^{2}y}{dx^{2}}+x\left(a+2bx^{q}\right)\frac{dy}{dx}+\left[c+dx^{2s}-b\left(1-a-q\right)x^{q}+b^{2}x^{2q}\right]y=0

In terms of the modified and unmodified Bessel functions of the first and second kinds, the general solution can be written as:

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

where c_1 and c_2 are constants and I_{\nu}(x) and K_{\nu}(x) are the modified and unmodified Bessel functions of the first and second kinds, respectively. The parameter \nu is related to the parameters a, b, q, and s in the equation.

I hope this helps clarify the form and solution of the general Bessel ODE that your professor referenced. Let me know if you have any further questions.