So I have a problem where I’m trying to solve some differential equations with split boundary conditions and have ran into some problems.

Firstly I define some parameters:

\[Lambda]i = 2; r0 = {-100, 500}; r1 = {300, 300}; u = {0, -1}; v = {1/Sqrt[2], 1/Sqrt[2]};

Then I start from a Lagrangian:

L = Sqrt[x'[s]^2 + y'[s]^2] + \[Lambda] (x'[s]*y''[s] -y'[s]*x''[s])^2/(x'[s]^2 + y'[s]^2)^3

and then generate two coupled differential equations:

eq1 = D[D[L, x'[s]], s] == D[D[L, x''[s]], {s, 2}] // FullSimplify
eq2 = D[D[L, y'[s]], s] == D[D[L, y''[s]], {s, 2}] // FullSimplify

which are very long and unpleasant to look at, but I’d like to be able to obtain a numerical solution with the following boundary conditions:

bc1 = x[0] == r0[[1]];
bc2 = x[1] == r0[[2]];
bc3 = y[0] == r1[[1]];
bc4 = y[1] == r1[[2]];
bc5 = x'[0] == u[[1]] *Sqrt[x'[0]^2 + y'[0]^2];
bc6 = x'[1] == v[[1]] *Sqrt[x'[1]^2 + y'[1]^2];
bc7 = y'[0] == u[[2]] * Sqrt[x'[0]^2 + y'[0]^2];
bc8 =  y'[1] == v[[2]]*Sqrt[x'[1]^2 + y'[1]^2];

However when I try using NDSolve:

NDSolve[{eq1 , eq2, bc1, bc2, bc3, bc4, bc5, bc6, bc7, bc8} /. \[Lambda] -> \[Lambda]i, {x, y}, {s, 0, 1}]

I get the errors:
“NDSolve::ntdvdae: Cannot solve to find an explicit formula for the derivatives. NDSolve will try solving the system as differential-algebraic equations.”

“NDSolve::bvdae: Differential-algebraic equations must be given as initial value problems.”

I tried to adapt an answer here and use parametricNDSolve to provide an initial value problem:

parSol2 = ParametricNDSolve[{eq1, eq2, bc1, bc3, bc5, bc7, x''[0] == xdd0, y''[0] == ydd0, x'''[0] == xddd0, y'''[0] == yddd0}, {x, y}, {s, 0, 1}, {xdd0, ydd0, xddd0, yddd0}]

which works but then trying to solve for the root doesn’t work, i.e.:

FindRoot[{bc2, bc4, bc6, bc8} /. {x -> x[xdd0, ydd0, xddd0, yddd0], y -> y[xdd0, ydd0, xddd0, yddd0]} /. parSol // Evaluate, {{xdd0, 0.1}, {ydd0, 0.1}, {xddd0, 0.1}, {yddd0, 0.1}}]

probably because I’m not providing a good initial guess (but I’m not sure how to go about improving this really with a 4 dimensional problem). Any suggestions?