Solving [h'(x)]^2 - h(x)·h''(x) + 1 = 0 for a hanging rope

[Semo727]

Hello!
I tried to prove, that ideal rope (see picture in attachment) has a shape of the function ch[x]. I finished with this equation

$$[h'(x)]^2-h(x)\cdot h''(x)+1=0$$

Yes, when you try function h[x]=ch[x], you get 0 on the left side, but I have no clue how to solve this equation (find the solution without knowing the solution). Even Mathematica has some problems, if I set boundary conditions. Could you please write how to solve this DE (providing it isn't too complicated, because I don't know much about solving nonlinear DE)

 

[BoTemp]

Guess and check is the only guaranteed way of finding a solution to any DE. You might be able to use a series expansion here (guess solution of form h(x) = sum(C_n * x^n) and plug in, but I'm not sure. That method is guaranteed to work for linear equations, non-linear it can get tricky.