# Calculus of variations functions

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

Solution:

(a) The associated Euler-Lagrange equation is:

( 1)

where

( 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).