Euler's Equation/Independent of x

Theorem
Let $y$ be a mapping.

Let $J$ be a functional such that:


 * $\displaystyle J \sqbrk y = \int_a^b \map F {x,y'} \rd x$

Then the corresponding Euler's Equation can be reduced to:


 * $F - y' F_{y'} = C$

where $C$ is an arbitrary constant.

Proof
Assume that:


 * $\displaystyle J \sqbrk y = \int_a^b \map F {y,y'} \rd x$

Then:

Multiply this equation by $y'$.

Then:

Integration yields the desired result.