Solving 3 Coupled ODEs with a Constraint

[vishal007win]

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?

 

[gtoguy97]

how?Also if i solve these two ODEs separately, is there any possibility that the solution of one ODE will satisfy the constraint equation or not? One possible approach is to integrate out one of the ODEs (say L2) and then substitute the resulting expression in L1 and C. This way you have two equations with two unknowns (x and λ). Then you can use methods from linear algebra to solve for x and λ. Once you have x and λ, you can use them to solve for y. This approach is known as the Lagrange multiplier method.