I've got two equations containing a function in z and 3 constants:
f(z) + 2AB + B - z^2 = 0 and
-Az^3 +(2A-B)Z^2 + Z(2f(Z) + 2B - C) - 2C
These equations have arisen from a fluid mechanics problem, but how would I solve them to find f(z)?
(when I do it I find f(z) to be a constant when it should be a function of z)
Firstly I'd like to say that I disagree with HallsofIvy with respect to the fact that for two such equations, if they are consistent, then you can find a single function f(z).
Both equations relate the function f(z) to polynomials in z and constants A,B and C. So the way I would go about it would be to rearrange your first equation explicitly in terms of f(z) then substitute for f(z) in equation 2. This will give you a cubic in z with coefficients related to the constants A,B and C. Since this equation must hold for all z then each of coefficients must equate to zero hence providing you with the values of the constants A,B and C which you plug back into equation 1 to give you your function f(z). Well, here's the snag - my quick attempt at doing this with your equations showed it to be impossible.
So either you need to check the derivation of your equations (you did say they are from fluid mechanics) or perhaps I misunderstood what you wrote.
Hope that's of some help to you!