# Problem with numerical solution of a system of ODEs

### Background

Good evening, I have a big problem with num. solution NDSolve of differential equation. To start with, the model:

There is a fast-moving rope in closed path. Where

$$T$$ is tension,

$$a$$ is angle beetwen rope and horizontal plane,

$$s$$ is curvilinear coordinates $$[0,1]$$ ($$0$$ - the beginning of the rope, $$1$$ - the ending of the rope).

$$Dr$$ drag coefficient

$$W$$ weight coefficient

And it gives the differential equations for $$T$$, $$a$$ and $$s$$.

$$\frac{d}{ds}(T(s)\sin\alpha(s))=W+Dr\sin\alpha(s)$$

$$\frac{d}{ds}(T(s)\cos\alpha(s))=Dr\cos\alpha(s)$$

D[T[s] Sin[a[s]], s] == W + Dr Sin[a[s]],
D[T[s] Cos[a[s]], s] == Dr Cos[a[s]],


Moreover, boundary conditions comes from equation for beginning and ending of the rope. $$x = x = 0, y = y = 0$$, where $$x[s]$$, $$y[s]$$ are coordinates of point on the rope with curve distance to beginning $$s$$. It means, that ending and beginning of the rope are in the same place. Differential equations for coodinates $$x[s],y[s]$$ are quite easy.

$$\frac{dy(s)}{ds}=\frac{\tan \alpha(s)}{\sqrt{1+\tan^2 \alpha(s)}}$$

$$\frac{dx(s)}{ds}=\frac{1}{\sqrt{1+\tan^2 \alpha(s)}}$$

(Tan[a[s]])/Sqrt[1 + Tan[a[s]]^2] == D[y[s], s],
1/Sqrt[1 + Tan[a[s]]^2] == D[x[s], s],
x == 0,
y == 0,
x == ϵ,
y == ϵ


I've solved it, but solution is unreal and contradicts boundary conditions. But Mathematica's solution in ParametricPlot looks like this: Fig.1 Obtained soltion

The rope should be closed, but it's not. And it should look like that: Fig.2 Shape of rope in dependence of $$\frac{Dr}{W}$$

x[s] =.