I have 3 nonlinear equations in 3 unknowns,

$\Large A=\frac{P_L}{1-c}\quad B=\frac{P_L}{1-ce^{-10k}}\quad C=\frac{P_L}{1-ce^{-20k}}$

where $A$, $B$, $C$ are known values and $P_L$, $c$ and $k$ are unknowns. I want to get them into one single equation in one of three unknowns ($P_L$, $c$ or $k$, does not matter) so I can use one of the root finding methods to determine that single unknown.

  • 1
    $\begingroup$ Put $x = e^{-10k}$ then between the first two equations you can show $c=\frac{(B-A)}{(Bx-A)}$. Use the second and third equations to write $\frac{B}{C}$ in terms of c and x. Now sub in your expression for c and multiply out to get a quadratic in x. $\endgroup$
    – Paul
    Dec 19 '20 at 21:32

You have $$c=1-\frac{P}{A}$$

$$ce^{-10k}=1-\frac{P}{B}$$ $$ce^{-20k}=1-\frac{P}{C}$$

Therefore $$(1-\frac{P}{B})^2=(1-\frac{P}{A})(1-\frac{P}{C})$$


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