Eliminate $z$ from these two hyperbolic functions.

Question

I was going through my textbook and was stuck at this one, i tried to do substitution but in vain, can you help me or give me a hint as to how to approach this problem?
Eliminate $z$ from these equations: $$ p\cdot \text{cosech} (z) + q\cdot \text{sech} (z) +r=0 $$ $$p'\cdot \text{cosech}(z) + q'\cdot \text{sech} (z) +r'=0 $$

Answer 1

You will get by definitions $$\frac{2p}{e^z-e^{-z}}+\frac{2q}{e^z+e^{-z}}+r=0$$ Substituting $$e^z=t$$ we get $$\frac{2pt}{t^2-1}+\frac{2qt}{t^2+1}+r=0$$ and you have to solve $$rt^4+t^3(2p-2q)+t(2p-2q)-r=0$$