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## Homework Statement

Minimize the functional: ∫

_{0}

^{1}dx y'

^{2}⋅ ∫

_{0}

^{1}dx(y(x)+1) with y(0)=0, y(1)=a

## Homework Equations

(1) δI=∫ dx [∂f/∂y δy +∂f/∂y' δy']

(2) δy'=d/dx(δy)

(3) ∫ dx ∂f/∂y' δy' = δy ∂f/∂y' |

_{0}

^{1}- ∫ dx d/dx(∂f/∂y') δy

where the first term goes to zero since there is no variation at the endpoints

## 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?