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Functional changing variables

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Changing variables in functionals

See attachment for full equation and answer the following questions:

a. Obtain the Euler-Lagrange equation corresponding to this functional.
b. By changing to a new independent variable u = exp(-2x), show that the functional S[y] is equivalent to the functional (see attachment)...
c. Determine the constants A and B, and show that the extremal value of the functional S[y] is equal to 2e/(e-1).

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Solution Summary

The 6-pages solution shows in detail how a change in variables can simplify the Euler-Lagrange equation and then goes on and find the extrema of the functional.

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See Attached

The functional is:
(1.1)
So the Lagrangian is:
(1.2)
And the functional attains its extrema when it satisfies Euler-Lagrange equation:
(1.3)
Then when we apply it to (1.2) we obtain the equation:

(1.4)

If we use the change of variables:
(1.5)
Note that
(1.6)
Furthermore:
(1.7)
Plugging all this into the ...

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