No , we should take total differential not partial ...
It is
Actually , we obtain immediately but in other situations , we should be careful that what we are doing is taking total differential .
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.
where and
and
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?
Just because a functional has an extremal does not mean that it is actually a minimum or maximum, just like how the first derivative equals zero does not imply you have a maximum there. In fact, the E-L equations can be thought very informally as an infinite dimensional analogue of a gradient.