1. Let I(u) = F(u'(x), u(x), x) dx. Find the first variation and Euler-Lagrange equation for each choice of F below. Then solve the equations to find a general solution for extremal of I(u). [Note: your solutions should have two constants of integration.] (a) F(p, u, x) = √1+p² U

1. Let I(u) = F(u'(x), u(x), x) dx. Find the first variation and Euler-Lagrange equation for each choice of F below. Then solve the equations to find a general solution for extremal of I(u). [Note: your solutions should have two constants of integration.] (a) F(p, u, x) = √1+p² U

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Please show a step-by-step solution. Do not skip steps, and explain your steps. Write it on paper, preferably. Please show the integrations by parts in detail. Make sure the work is clear.

1. Let I(u) = F(u'(x), u(x), x) dx. Find the first variation and Euler-Lagrange equations
for each choice of F below. Then solve the equations to find a general solution for extremals
of I(u).
[Note: your solutions should have two constants of integration.]
√1+p²
U
(a) F(p, u, x) =
(b) F(p, u, x) =
p² + u²
(c) F(p, u, x) = /u² - p²

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Please don't provide hand written solution
Your company are designing metal trays. They will be made out of a sheet of metalwhich has a perimeter of 80 cm. The box is made by removing squares of side y = 2cm from each corner of the rectangle and then folding in the sides. A net is shownbelow.● Your company wants to maximise the capacity of these trays and wants you toinvestigate what value of x will give you the maximum volume.● Investigate how changing the size of the square at each corner would affect thecapacity of the tray and investigate the value of y which would give youmaximum volume.