1. The problem statement, all variables and given/known data Minimize the functional: ∫01 dx y'2⋅ ∫01 dx(y(x)+1) with y(0)=0, y(1)=a 2. Relevant equations (1) δI=∫ dx [∂f/∂y δy +∂f/∂y' δy'] (2) δy'=d/dx(δy) (3) ∫ dx ∂f/∂y' δy' = δy ∂f/∂y' |01 - ∫ dx d/dx(∂f/∂y') δy where the first term goes to zero since there is no variation at the endpoints 3. The attempt at a solution Using (1) to get δI= ∫dx 2y'δy' ∫dx(y+1) + ∫dx y'2 ∫dx δy Using (2) and (3) to get δI= ∫dx y'2 ∫dx δy - ∫dx d/dx(2y')δy ∫dx(y+1) = 0 I then tried various ways of manipulating this equation to something that I could minimize trying to isolate δy but was unable to do so since it was always within an integral. I tried to simplify d/dx(2 y') to 2y'' to see if that helped but could not find any benefit to doing so. I also thought about using an integral by parts again but couldn't see where to do so. Using the values at the end points seems like something I should be doing but I don't know how I would. Any advice on how to proceed?