So I decided to use the Euler Lagrange equations:Find solutions of the Euler-Lagrange equations for critical points of the following functionals with the given endpoint conditions.
Inputting this into the Euler-Lagrange equations gives:
So integrating both sides w.r.t x gives where is a positive constant.
From here I was expecting to use the conditions on y to work out what A is. Unfortunately this doesn't work! y can't equal a constant and yet change from 1 to 2 as described in the conditions.
What's going wrong?