real analysis - $J[y]=int_a^bF(x,y,y')dx$ with constraint and free boundary

Suppose the variation problem
$$J[y]=\int_a^bF(x,y,y')dx$$
with free boundary and constraint $\int_a^bG(x,y,y')=l$, how can formulate the corresponding Euler-Lagrange equation?

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For fixed boundary i.e. $f(a)=A, f(b)=B$, the Euler-Lagrange equation is
$$F_y-\frac{d}{dx}F_{y'}+(G_y-\frac{d}{dx}G_{y'})=0$$