## finding extremal -no boundary conditions specified, exam on tuesday!

I think this is going to come up on my exam so I'm worried as I can't do it...

a) If $F = y^2 - 2xy - y'2, y(0) = 1, y(\pi/2) = 0$ show that the extremal is $y_0(x) = x - \frac{\pi}{2} sin x + cos x.$.

b) Find the extremal in which $y(\pi/2)$ is not specified.

I've can do part a fine, and part b is:

Transversality condition: $Fy'$(at x =b) $= 0.$

Hence $-2y_0' (\pi/2) = 0$

$y_0(x) = cos x + B sin x + x$

$y_0'(x) = -sin x + B cos x + 1$

$0 = -1 + 1$

Hence there are infinitely many extremals.

But for a similar question:

a) If $F = y'^2 + 2y -y^2, y(0) = 2, y(\pi/2) = 0$ show that the extremal is $y_0(x) = cos x - sin x + 1.$.

b) Write down the transversality condition if both $y(0)$ and $y(\pi/2)$ are not specified.

For part b I don't understand how to do this without any boundary conditions at all...could someone please explain how?