I have a question that "feels" basic/stupid, but I've been really struggling with it. The basic question is: is there an easy way to determine t(x)$t(x)$ from x(t)$x(t)$ or dx/dt$\frac{dx}{dt}$?
To explain further, I have a differential-algebraic system of equations I'm trying to solve. The system I'm solving is technically a 1D partial-differential equation, but all of the academic papers I've come across (at least 15) eliminate the spatial dimension by numerically integrating it, and this t(x) term shows up in the integral. However, none of them give any details on HOW they do that - they just say "numerical methods were used, and it was easily solved." See below for a representation of the full equation set:
$$\frac{d(F)}{dt} = C - E(t) $$ $$\frac{d(x)}{dt} = \sqrt{g*H(t)}$$ $$H(t) = \frac{F(t)}{\pi*x(t)^2}$$ $$E(t) = Const*\int\limits_0^{x(t)}\frac{2*\pi*x(t)}{t(x)} dx$$
I've managed to calculate t(x) via brute force with a coding algorithm, but I feel like there should be an easier way to find it either analytically or numerically. My approach is not very robust - especially for small values of x/t (x and t both technically start at 0, where everything goes to infinity - so I add a small offset to prevent divide-by-zero issues). All of the other equations/math are very straightforward and make sense to me - this one term is just tripping me up!
