I was trying to solve a particular problem and I have reduced it down to an integral/differential equation that I do not know how to solve. Here it is in fully explicit form:
$$\int\limits_{t-\frac{d}{2}}^{\frac{d}{2} - t}{\sqrt{1+\bigg(\frac{\partial y}{\partial x}\bigg)^2} dx = d}$$
I am looking for $y(x, t)$ with these boundary conditions:
$$y\left(t-\frac{d}{2}, t\right) = y\left(\frac{d}{2}-t, t\right) = h$$
$h$ and $d$ are positive real numbers.
I don't know if it'll help but from the context of the problem I know that the $y$ at any constant $t$ is an even function of $x.$
Numerical solutions would also do if it's very hard to solve analytically.
This doesn't have a unique solution, since you can have two ropes of equal length span the same two points, yet follow different paths.
Imagine one ropes that's like a "smile" between the two points and another like a "frown", they are reflections about the line directly from one anchor to the next.
The only case where only one solution exists is when it's the distance between the anchors. No solution exists if the prescribed distance is shorter than that.
A case where there is one solution to such a problem is the minimum length case, or the case when you seek a minimum length for two boundary conditions under the influence of weight, this is called a catenary.