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Question 6.9 Taylor: Classical Mechanics

  1. Jan 3, 2017 #1
    1. The problem statement, all variables and given/known data
    Hello, I solved others but not 6.9:
    Find the equation of the path joining the origin O to point P(1,1) in the xy plane that makes the integral ∫(y'2 +yy' + y2) dx stationary.
    ∫ from O to P. y' = dy/dx

    2. Relevant equations
    I need use ∂f/∂y = d/dx (∂f/∂y') (euler-lagrange equation) with f = y'2 +yy' + y2
    ∂f/∂y = y' + 2y
    ∂f/∂y' = 2y' + y

    and d/dx (∂f/∂y') = 2 y'' + y', go to euler-lagrange equation we get:
    y' + 2y = 2 y'' + y'
    this is equivalent to:
    y'' = y (eq 1)

    y = ex is solution of eq. 1, but it don't fit (0,0) and (1,1)
    I see the solution at final of the book: y = senh(x)/senh(1)
    Ok, is solution of eq. 1 and fit points (0,0) and (1,1).

    There are any thing more? I feel I dont get good answer without look at the final of the book.
    Thanks! Luiz

     
  2. jcsd
  3. Jan 3, 2017 #2

    vela

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    You have a second-order differential equation, so you should have two solutions. You found one. What's the other one? The general solution will be a linear combination of the two solutions.
     
  4. Jan 3, 2017 #3
    Ok, 1st is y = A1 ex 2nd is y = A2 e-x and general solution is y = A1 ex + A2 e-x wich go to y = senh(x)/senh(1).
    Ok, remember this (two solutions) is fine way to get the right answer... :)
     
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