# How to solve this complex nonlinear system of equations? (comes from solving catenary equations)

I am working on solving a system of equations where I simplified to two equations and two unknowns. The two equations are nonlinear and complicated, and I am unsure how to solve it. Having done equations with easier, well-made algebraic solutions in college, I don't know how to work with more complicated equations. Here are the equations, please help me with this problem:

$$\left(A\times\sinh^{-1}\dfrac{\dfrac{L}{2}+\sinh\dfrac{a}{A}}{A}-a\right)L=27.625\left(L-\sinh\dfrac{a}{A}\right)+25.875A\\\dfrac{L}{A}=\sinh\dfrac{27.625+a}{A}-\sinh\dfrac{a}{A}$$ where $L$ is a given.

Thank you!

• What are you trying to solve for? I'd strong suspect that it's unsolveable for either $a$ or $A$ – Michael Stachowsky Apr 3 '18 at 15:15
• I am trying to solve for $A$. Although solving for either should work to solve the other for this system of equations. – Jay Yang Apr 3 '18 at 15:17
• It's highly unlikely that you can solve this explicitly. You will always end up with $A$ in a hyperbolic function, no matter what kind of manipulation you try to do. At best, I suspect you can solve it numerically. however, I can't prove that, so if someone else comes along and solves it I'll retract – Michael Stachowsky Apr 3 '18 at 15:20
• I don't have access (or the skill) to use advanced software for getting a numerical answer to a question like this one. Could someone help lead me to such resources? I tried Wolfram Alpha (all I know) and it actually timed out. – Jay Yang Apr 3 '18 at 15:32
• You certainly have access. Download GNU Octave. It's a free/open source version of MATLAB, and can handle this computation. In terms of the skill, that's where you'll need to learn to use it, and that may take time. For what it's worth, fzero is the function you should be starting with, or fminsearch. – Michael Stachowsky Apr 3 '18 at 15:34

The second equation $$\dfrac{L}{A}=\sinh\left(\dfrac{t+a}{A}\right)-\sinh\left(\dfrac{a}{A}\right)\qquad \text{where} \qquad t=27.625$$ can rewrite as $$\dfrac{L}{A}=2\sinh\left(\dfrac{t}{A}\right)\cosh\left(\dfrac{t+2a}{A}\right)$$ from which we can eliminate $a$ to get two solutions $$a_{\pm}=-\frac{1}{2} \left(A \cosh ^{-1}\left(\frac{L }{2 A}\text{csch}\left(\frac{t}{A}\right)\right)\pm t\right)$$ For each case, replace this nice expression in the first equation which is now function of $A$ only. Plot them to locate more or less where is the solution and use Newton method with numerical derivatives (except if you enjoy nightmares).