I have a system of equations that I think should be rather simple to solve, but I can't seem to get to the answer:

f(r) = A Sin(j*r)

g(r) = B e^(k*r) + C e^(-k*r)

h(r) = D e^(-l*r)

I have 4 boundary conditions:

f(r)=g(r) at r=a

f'(r)=g'(r) at r=a

g(r)=h(r) at r=a+p

g'(r) =h'(r) at r=a+p

j, k, and l are functions of E that I want to find the roots of. p and a are constants, but I want to be able to solve these equations for any value of a and p

With this information I should be able to eliminate A, B, C, and D and be able to calculate the values of E which give the zeros.

I can simplify the system to:

-j*((( B e^(k*a) + C e^(-k*a))/( B e^(k*a) + C e^(-k*a)))*(cot j*a) = l*(( B e^(k*(a+p)) + C e^(-k*(a+p))/( B e^(k*(a+p)) + C e^(-k*(a+p))

but I can't seem to get rid of B and C. When I look at it written out it looks like I should easily be able to remove B and C, but it never works. I feel like there is some rational or exponential rule that I can't remember that would help me out.

Thanks a lot and sorry if the notation is confusing.