A functional depending upon x and y'(x)

[mistereko]

Homework Statement

S[y] = \int²₁dx ln(1 + xⁿy'), 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/xⁿ


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] = \int²₁dx (xⁿh(x)'/(1 + xⁿy(x)')ε - ε/2 \int²₁dx x²ⁿ(h(x)')²/(1+ xⁿy(x)')² + O(ε³)

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! :)

 

[mistereko]

Anyone?