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
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 8CR
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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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