# NDSolve error for system of two non-linear equations

I'm trying to solve this system of non-linear ODE's $$\left\{y'(s) \left(x''(s) y'(s)-x'(s) y''(s)\right)=0,\mu x'(s) \left(x'(s) y''(s)-x''(s) y'(s)\right)=g \lambda \left(x'(s)^2+y'(s)^2\right)^{3/2}\right\}$$ where for instance $$g=\mu=\lambda=1$$

The code with set boundary conditions is

NDSolveValue[{-2 Derivative[y][s] (x^\[Prime]\[Prime])[
s] + Derivative[x][s] (y^\[Prime]\[Prime])[s] ==
0, (-2 Derivative[x][s] Derivative[y][s] (
x^\[Prime]\[Prime])[s] +
2 Derivative[x][s]^2 (y^\[Prime]\[Prime])[s] +
Derivative[y][s] (y^\[Prime]\[Prime])[s]) ==
2 (Derivative[x][s]^2 + Derivative[y][s])^(3/2),
DirichletCondition[x[s] == s/2, True],
DirichletCondition[x[s] == 1, True]}, {x[s], y[s]}, {s, 0, 1}]


I get a NDSolveValue::femnlmdor error: I don't really understand where this error is coming from, what's wrong with my system begin second order so long as it has sufficient boundary conditions. Any insight would be appreciated.

• could you explain better what the BC are? You have 1D problem. What are the dependent variables values are boundaries? – Nasser Apr 13 at 4:49
• There is one independent variable s and two dependant x and y. The boundary conditions are specified at x==0, x==1/2, y==1 and y==1 – user2757771 Apr 13 at 4:51
• if you do not solve using FEM, the error is Cannot solve to find an explicit formula for the derivatives. NDSolve will try solving the system as differential-algebraic equations. It is better to write your code using separate statements so easy to read. mu = 1; g = 1; lambda = 1; ode1 = y'[s] (x''[s]*y'[s] - x'[s]*y''[s]) == 0; ode2 = mu*x'[s]*(x'[s]*y''[s] - x''[s]*y'[s]) == g*lambda*(x'[s]^2 + y'[s]^2)^(3/2); bc = {x == 0, x == 1/2, y == 1 , y == 1}; NDSolveValue[{ode1, ode2, bc}, {x[s], y[s]}, {s, 0, 1}] – Nasser Apr 13 at 5:05
• y'[s] (x''[s]*y'[s] - x'[s]*y''[s]) == 0 has two possible solutions. The first is y[s] constant, which when substituted into the second equation also yields x[s] constant. The second is (x''[s]*y'[s] - x'[s]*y''[s]) == 0, which when substituted into the second equations again yields both x[s] and y[s] constant. – bbgodfrey Apr 13 at 11:47
• If you want to make use of the FEM then you'd need to rewrite this equation in Inactive form. See this message page; FEMDocumentation/ref/message/InitializePDECoefficients/femnlmdor – user21 Apr 14 at 7:03