May be this helps:
I don't know if this is true:
Then we may integrate two times by parts.
Hello, was wondering if you could help me with a question im stuck on...
The functional I is defined by
I[y] = integral from 1 to 0 of (1/2 y''^2) dx + (y(1))^2 + y'(1),
where y(0) = y'(0) = 0 and y belongs to C^4[0,1].
Show from first principles that a extremum y0(x) of I satisfies the euler equation y0'''' = 0
together with the transversality conditions y0''(1) = -1, y0'''(1) = 2y0(1).
show that y0= -1/10x^2(2+x).
Thanks : ) x