Could anyone help me find the extremals of I[y]=\int_0^1(y')^2 \mathrm dx+\{y(1)\}^2 subject to $y(0)=1$
Most crucially I can't work out how to find the boundary $x=1$. I'm trying to go back to first principles and letting $y \rightarrow y+\alpha \eta$. Here the normal step would be to differentiate and set $\alpha=0$ and hence derive the Euler-Lagrange equation.
If anyone can explain to me how to deal with this case i'd be very grateful!