3. Let p: R → R be a differentiable function. Prove that the equation Uų = p(u)u, t > 0 has a solution satisfying the functional relation u = entiable function. In particular find such solutions for the following equations: f(x+p(u)t), where f is a differ- (a) ut = (b) u = ux; (c) u = u sin(u)uz. (Note: u = = u(x, t), so the chain rule applies to p(u) as well.)
3. Let p: R → R be a differentiable function. Prove that the equation Uų = p(u)u, t > 0 has a solution satisfying the functional relation u = entiable function. In particular find such solutions for the following equations: f(x+p(u)t), where f is a differ- (a) ut = (b) u = ux; (c) u = u sin(u)uz. (Note: u = = u(x, t), so the chain rule applies to p(u) as well.)
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
Transcribed Image Text:3. Let p : R R be a differentiable function. Prove that the equation
Uų = p(u)u
t > 0
has a solution satisfying the functional relation u = f(x + p(u)t), wheref is a differ-
entiable function. In particular find such solutions for the following equations:
(a) ut = kuz;
(b) ut = uU;
(c) ų = usin(u)uz-
(Note: u = u(x,t), so the chain rule applies to p(u) as well.)
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