- #1

mistereko

- 26

- 0

## Homework Statement

S[y] = [itex]\int[/itex]

^{2}

_{1}dx ln(1 + x

^{n}y'), y(1) = 1, y(2) = 2

^{1-n}

where n > 1 is a constant integer, and y is a continuously diﬀerentiable function

for 1 ≤ x ≤ 2. Let h be a continuously diﬀerentiable function for 1 ≤ x ≤ 2 and ε

a constant. Let ∆ = S[y + εh] − S[y].

Show that if h(1) = h(2) = 0, the term O(ε) in this expansion vanishes if

y′(x) satisﬁes the equation

dy/dx = 1/c -1/x

^{n}

where c is a constant.

Solve this equation to show that the stationary path is

y(x) = (n-2)(2

^{n-1}-1)(1-x)/2

^{n-1}(n-1) + 1/(n-1)x

^{n-1}+ n-2/n-1

## Homework Equations

## The Attempt at a Solution

Right, I've found S[y+ εh] -s[y] = [itex]\int[/itex]

^{2}

_{1}dx (x

^{n}h(x)'/(1 + x

^{n}y(x)')ε - ε/2 [itex]\int[/itex]

^{2}

_{1}dx x

^{2n}(h(x)')

^{2}/(1+ x

^{n}y(x)')

^{2}+ O(ε

^{3})

I'm aslo pretty sure I've proved the O(ε) vanishes. I did this substituting y(x) = (n-2)(2

^{n-1}-1)(1-x)/2

^{n-1}(n-1) + 1/(n-1)x

^{n-1}+ n-2/n-1 into ε term and got it to equal 0. I don't know how to find y(x) though. Through integration i got y(x) = 1/c - (1

^{-n+1})/(-n+1)

I don't know how to progress from here or if I'm doing it correctly. Some guidance would be great! :)