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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 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).

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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 ...

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