Extremals - Euler - Lagrange Equation
I don't know if someone will be able to help me with this, but I'm going to post it anyway. I'm studying a worked example, but I'm confused about a few things and was hoping someone could help me understand some steps.
Find the extremal for the following fixed-end problem
Solution (I'm going to number the steps here.
(1) Let . The Euler-Lagrange equation is
(2) which implies
(3) so where C is a constant.
(4) Then where
(5) Using we have
(6) Solving this linear system yields
(7) So the extremal is
Firstly, I don't understand how this goes from (2) to (3) i.e where , and from (3) to (4).
Can anyone please explain it?
Another example I have (which I won't post the whole thing for) is
Find the extremal of
(9) The Euler-Lagrange equation is
(10) which implieds
(11) This implies
again, in this example I don't understand how 11 was arrived at from 10. (Note the extremal solution to this second example is