First integral of Euler–Poisson when $f$ omits $y$

If the integrand $f$ does not depend on $y$, the Euler Lagrange's equation of the functional $I[y(x)]=\int_{x_1}^{x_2}f(x,y',y")dx$ under the conditions that both $y$ and $y'$ are prescribed at the end points has the first integral as: