I have been analysing a problem involving catenaries and I have derived an equation of the following form:
$$\big(Ax + B\big)\sinh\left(\frac{k}{Ax + B}\right) = Cx + D$$
In this equation, $x$ is the only variable. Everything else is a constant.
I have been trying to solve this equation but haven't been able to. Could you please provide pointers as to how I might be able to solve it? Thanks!
This problem will require numerical methods or approximations
You want to find the zeros of function $$f(x)=\big(Ax + B\big)\sinh\left(\frac{k}{Ax + B}\right) -(Cx + D)$$ In order to make it simpler, let $$t=\frac{k}{Ax + B} \qquad a=\frac{AD-BC}{A k} \qquad b=\frac CA$$ and consider the function $$g(t)=\sinh(t)-(at+b)$$ which is much simpler.
Graphing, you have an estimate of the root(s) and you can polish the solution using Newton method.
