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    Calculus of variations functions

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    Hi, please see the attached question which I am having trouble with and need help with.

    I have done part a) and part b)
    the E-L equation for part a) I got to be 2y''sinhx - 2y'coshx + ysinh^3(x) = 0

    and for part b) the new variable I have is u = cosh x and with new limits u1 = cosh a and
    u2 = cosh b
    and for part c) I've managed to get as far as the E-L equation being 2y''- y = 0 with a general solution of

    y(u)=Acos(u/sqrt2) + Bsin(u/sqrt2)

    which in terms of x is y(x) = Acos(coshx/sqrt2) + Bsin(coshx/sqrt2)

    I would really appreciate some help on the rest of part c) and part d) and if you could also check that what I have for parts a), b) and my partial part c) is correct.

    Thank you.

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    Solution Preview

    see attachment


    (a) The associated Euler-Lagrange equation is:
    ( 1)
    ( 2)
    By differentiating, we get:
    ( 3)
    ( 4)
    The equation Euler-Lagrange becomes:
    ( 5)
    or, by multiplying with - sinh2(x):
    ( 6)
    (b) Let's consider a variable change of form y = y(u) where
    ( 7)
    The given functional changes as follows:
    ( 8)
    ( 9)
    But ...

    Solution Summary

    A problem of calculus of variations is solved using a variable change which simplifies the differential equation obtained by applying Euler-Lagrange equation for the given functional which has to be minimized (or maximized).