Hello!

Could someone please help me?

I have an equation

and need to find its roots. Here

and

are Bessel functions (well, a Bessel and Neumann functions, respectively). It's obvious that the equation has infinitely many roots, and the plot of this function (in case of

) is below the text of this message. Ho do I get the roots? I need a fast and accurate method. Basic numerical methods don't work here.

What have I tried? Well, I use Mathematica with its nice command "FindRoot", bet there are some difficulties with its use in this case (especially when x is small). Second, the series of these functions are too complex to analyse them. And third, as I said, standard numerical methods (for example, Newton's method) can't help here. I haven't met this equation in any book/textbook.

Since

it makes sence to get the roots from right to left and stop the process at a certain point.

I've found a quite similar equation

here (the last one in the list). One can obtain a lot of roots in a couple of seconds using this calculator. The good news for me is that Mathematica is used there. How do they do that?