Euler–Poisson with $f_y=0$ integrates once: $f_{y'}-\frac d{dx}f_{y"}=$ const

If the integrand $f$ does not depend on $y$, the Euler–Lagrange equation of the functional $I=\int_{x_1}^{x_2}f(x,y,y',y")\,dx$, with $y$ and $y'$ prescribed at the end points, has the first integral: