Bessel Funcs: Proving J1/uJ0 + K1/wK0 & k12J1/uJ0 + k22K1/wK0

[Gogsey]

Hi,

This is a question about modes in step index fibers, however its just the math in the following equations that I'm having trouble with, so you don't need to know the question.
basically we have the following 2 equations:

J₁/uJ₀ + K₁/wK₀ = 0

k₁²J₁/uJ₀ + k₂²K₁/wK₀ = 0

where J_n is a first order Bessel function, and K_n is a second order Bessel function.

We have to prove this but I have no idea how to do it. I've looked up various Bessel functions for all the J_n and K_n values so I could input them into the equation, but its so complicated and there's multiple expressions for them, I don't think that's what we are supposed to do.

 

[NateD]

Any help would be greatly appreciated!The key to solving this is to recognize that the two equations are linear equations with two unknowns, J1/uJ0 and K1/wK0. This means that we can solve for both unknowns using the same technique.To do so, we can use elimination, which is a method of solving linear equations by adding or subtracting them in order to eliminate one of the unknowns. For example, if we multiply the first equation by k22 and the second equation by uJ0 and then add the two equations together, the k22 term will cancel out and we will have an equation with only one unknown (K1/wK0). We can then solve for this unknown and substitute it back into either of the original equations to solve for the other unknown.Once you have done this, you should be able to prove that the two equations are equivalent.