1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A functional depending upon x and y'(x)

  1. Mar 15, 2012 #1
    1. The problem statement, all variables and given/known data

    S[y] = [itex]\int[/itex]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] = [itex]\int[/itex]21dx (xnh(x)'/(1 + xny(x)')ε - ε/2 [itex]\int[/itex]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. jcsd
  3. Mar 20, 2012 #2
    Anyone?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: A functional depending upon x and y'(x)
Loading...