while solving Lagrangian of a system to derive equations of motion in presence of a constraint, I have finally landed down to a system of 3 coupled ODEs , where i have two variables(x and y) and 1 Lagrange multiplier. ODEs are of order 4,3 and 1 respectively. L1(x,y)=lambda L2(x,y)=0 C(x,y)=0 where L1 and L2 are odes and C is the constraint Now how to solve such a system. Any suggestions? My attempt was: after deriving ODEs L1 and L2, i applied constraint and got these 2 ODEs in one variable by replacing another one, but i am still confused, these two ODEs will give the same solution? or i need to manipulate between these two as well?