# A functional depending upon x and y'(x)

1. Mar 15, 2012

### mistereko

1. The problem statement, all variables and given/known data

S[y] = $\int$21dx ln(1 + xny'), y(1) = 1, y(2) = 21-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)(2n-1 -1)(1-x)/2n-1(n-1) + 1/(n-1)xn-1 + n-2/n-1

2. Relevant equations

3. The attempt at a solution

Right, I've found S[y+ εh] -s[y] = $\int$21dx (xnh(x)'/(1 + xny(x)')ε - ε/2 $\int$21dx 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)(2n-1 -1)(1-x)/2n-1(n-1) + 1/(n-1)xn-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!! :)

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Mar 20, 2012

Anyone?