Solving differential equation from variational principle

I have the following differential equation which I obtained from Euler-Lagrange

variational principle

I also have two boundary conditions: and

where , and are known

numbers.

I assume that I should integrate once to get

where is a function of . I would like to get

as a function of but the problem is that I don't know the

form of .

My question, how to solve this equation to get as a function

of . Is it possible to guess the form of , e.g.

from the bounday conditions.

Re: Solving differential equation from variational principle

Since y is a function of x, then f(y) is function of x. It's derivative is 0, hence f(y)=constant.

Then, solve dy/dx=c*sqrt(y), which is a separable ODE.