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.] (a) F(p, u, x) = 1+p² U (b) F(p, u, x) = (c) F(p, u, x) = ² - /p² √p² + u²
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.] (a) F(p, u, x) = 1+p² U (b) F(p, u, x) = (c) F(p, u, x) = ² - /p² √p² + u²
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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Please show a step-by-step solution. Do not skip steps, and explain your steps. Write it on paper, preferably. Make sure the work is clear.
Please find the first variation and then use the Euler Lagrange. Solve for question c.
![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.]
(a) F(p, u, x)
=
1+p²
U
(b) F(p, u, x) =
(c) F(p, u, x) = ² - /p²
√p² + u²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cb378ba-be87-4838-bd31-84fa6b2aaf1f%2F0864fb31-9767-48ff-9ef3-95c896c5247b%2F9mrae68_processed.png&w=3840&q=75)
Transcribed Image Text: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.]
(a) F(p, u, x)
=
1+p²
U
(b) F(p, u, x) =
(c) F(p, u, x) = ² - /p²
√p² + u²
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