Bessel function with differential equations

How do we use the matlab command besselj(0,x) for differential functions?

[Torsten]

So what is your task ? Finding zeros of Bessel functions ? differentiate Bessel functions ? plot them ?

Please state your question clearly.

[Howie]

If I have a differential equation how do I use the besselj(0,x) command to solve?

[Torsten]

If your differential equation (or its solution) contains the Bessel function of the first kind of order 0 (usually written as J_0(x)), you can get its value y at x by using

y = besselj(0,x)

[Howie]

Can you show me an example of you using it with a differentiable equation?

[Torsten]

fun = @(t,y) besselj(0,y);
y0 = 1.0;
tspan = [0,1];
[T,Y] = ode45(fun,tspan,y0);
plot(T,Y)

[Howie]

What's the differential equation in this case?

[Torsten]

y'(t) = J_0(y(t))
y(0) = 1

[Howie]

Thank you!

[John D'Errico]

I'm reminded of an old story my advisor liked to tell about Bessel functions. Thus, a world famous professor is visting the local school, and has been invited to give a presentation on his work to the students and faculty. He proceeds to cover one blackboard after another with dense mathematics, hardly stopping for questions. At one point however, seeing that many seem to be nodding off, he turns around and asks if there are any students in the audience who have never seen a Bessel function before. One student gathers the courage to meekly raise his or her hand, admitting the fact. The professor nods, turns back to the blackboard and points, well, there is one now. Then he proceeeds with his lecture.

Anyway, your question does not make complete sense, as it can be interpreted in many ways. For example, a Bessel function can be viewed as the solution to a specific differential equation. For example:

https://en.wikipedia.org/wiki/Bessel_function

https://math.libretexts.org/Bookshelves/Differential_Equations/Book%3A_Partial_Differential_Equations_(Walet)/10%3A_Bessel_Functions_and_Two-Dimensional_Problems/10.02%3A_Bessel’s_Equation

Or, you can use Bessel functions to solve a differential equation, thus forming the solution as a series of Bessel functions. Again, look at the wikipedia link, where they point out some of the problems where one finds Bessel functions naturally arise.

But asking how you use the function bessel itself? It is not something you can just pass a differential equation to, and expect it to return a solution (as you would with dsolve), any more than you can pass a differential equation to the sin function, and expect a Fourier series solution to be returned.

Bessel functions have the nice property that they are sort of Like the sine and cosine family of functions, or even the family of polynomials. That is, you can expand a result as an infinite series of those functions, WHERE that is appropriate.

I'm still not sure what you are asking here.