I need to solve the Euler equation to make the following integral stationary:

so I happily assume and find the partial derivative F wrt y' and F wrt y, put both into the Euler equation:

Route 1: simple rearrangement yield:

Route 2: intergate both side by x

Then I'm stuck in both.... there is probably a very simple way of getting the solution out, but my brain just can't turn the concer~ or I've already made a mistake early in my calculations?

Thanks for all the help!