Probelm 3) Consider the ordinary differential equation a' = f(x,t), where f : R → R is of class C'. Suppose that f(x,t+T) = f(x,t) for allte R. Moreover, suppose that there exist a and b with a < b such that f(a,t) > 0 and f(b,t) < 0 for all t e R or vice versa. Prove that a' = f(x,t) has a T-periodic solution x(t) such that a < x(t) < b for all t e R. What can you say about the ordinary differential equation a' = x² – 1– cos(t).
Probelm 3) Consider the ordinary differential equation a' = f(x,t), where f : R → R is of class C'. Suppose that f(x,t+T) = f(x,t) for allte R. Moreover, suppose that there exist a and b with a < b such that f(a,t) > 0 and f(b,t) < 0 for all t e R or vice versa. Prove that a' = f(x,t) has a T-periodic solution x(t) such that a < x(t) < b for all t e R. What can you say about the ordinary differential equation a' = x² – 1– cos(t).
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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Transcribed Image Text:Probelm 3) Consider the ordinary differential equation a' = f(x,t), where f : R → R is of class
Cl. Suppose that f(x,t+T) = f(x,t) for allte R. Moreover, suppose that there exist a and b
with a < b such that f(a,t) > 0 and f(b,t) < 0 for all t ER or vice versa. Prove that x' = f(x,t)
has a T-periodic solution x(t) such that a < x(t) < b for all t e R. What can you say about the
ordinary differential equation a' = x² – 1 – cos(t).
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