Solve constants in Coffin-Manson Equation

Question

Solve for $b, c, x, y$, given the following: $$\begin{cases} x\cdot1000^b + y\cdot1000^c = .0068\\ x\cdot2000^b + y\cdot2000^c = .0058\\ x\cdot4000^b + y\cdot4000^c = .0050\\ x\cdot6000^b + y\cdot6000^c = .0045 \end{cases}$$ Note: this is a simplified Coffin-Manson equation for fatigue for copper alloy C18150.

Answer 1

In further questions, please use MathJax to enter formulae like in the current edit.

Doesn't seem to reduce to a linear system... One can proceed by eliminating variables, but in the end (or in the middle, if there was say $5000$ instead of $4000$...) you will always get transcendental powers and have to resort to numerical methods.

What are the constraints on $b,c,x,y$? I've eliminated $x$ and $b$ by hand, then there was a quadratic equation for $z = .0058 - 2000^c y$ in terms of functions of $c$ (the fact that $4000/2000 = 2000/1000$ helped). Using Mathematica and Newton's method, I found a root value for $c$ and then back-substituted to other equations. Different signs for $z$ produce these values for $x,b,y,c$:

$(-0.3208412116266972, -0.9794915849630274, 0.04318887719378185, -0.2599575619044979)$

$(-0.9678214752352529, -0.26134267209576073, 1.0000202057445287, -0.26002310279335056)$

Are these inane for the problem? They seem to be close to solutions, and in principle, you can use a numerical method to get a better approximation. (For me, Mathematica's FindRoot fails that.)