Solving non-linear coupled ODE with constraint using NDSolve

I am trying to use NDSolve in Mathematica to solve the following coupled ODEs from a paper I am trying to understand. I have the following 5 differential equations:

eqns = {x'[s] == Cos[θ[s]],
y'[s] == Sin[θ[s]],
θ''[s] == nx[s]*Sin[θ[s]] - ny[s] Cos[θ[s]],
nx'[s] == -(γ/r) Sin[θ[s]],
ny'[s] == (γ/r) Cos[θ[s]]};

And i have to solve them with 8 boundary conditions

bcs = {x == 0,
y == 0,
θ == 0,
x[D] == r Sin[β],
θ'[D] == 0,
nx[D] == -γ Sin[β],
ny[D] == γ*Sin[β],
θ[D] == θY - β};

and a constraint:

constraint = r^2 (β - 1/2 Sin[2 β]) + 2 x[D] y[D] -
2 \!$$\*SubsuperscriptBox[\(∫$$,  $$0$$, $$D$$]$$y[s] Cos[θ[s]] \[DifferentialD]s$$\);

I am trying to learn how to use shooting method which is what is apparently used in their work to solve this system. Can anyone suggest a way to do that or is there an alternate way of solving these kind of coupled non-linear systems with constraint? If you have a simpler example where shooting method is used with constraints, would also be helpful.

• Googling shooting method mathematica returns nothing then, hmm .. ?! – Sektor May 17 '15 at 14:17
• @Sektor it did but i couldn't find one with a constraint such as this one, with an integral. And i am still a beginner in mathematica. – sgp May 17 '15 at 14:25
• Do you know how to solve even a basic problem using the shooting method ? – Sektor May 17 '15 at 16:26