Euler's Equation/Independent of y

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:


 * $F_{y'} = C$

where $C$ is an arbitrary constant.

Proof
Assume that:


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

Euler's equation for $J$ is:


 * $\dfrac \d {\d x} F_{y'} = 0$

Integration yields:


 * $F_{y'} = C$