22.Prove that for the system dxdt=F(x,y),dydt=G(x,y) there is at most one trajectory passing through a given point (x0, y0). Hint: Let C0 be the trajectory generated by the solution x = ϕ0(t), y = ψ0(t), with ϕ0(t0) = x0, ψ0(t0) = y0, and let C1 be the trajectory generated by the solution x = ϕ1(t), y = ψ1(t), with ϕ1(t1) = x0, ψ1(t1) = y0. Use the fact that the system is autonomous, and also the existence and uniqueness theorem, to show that C0 and C1 are the same.
22.Prove that for the system dxdt=F(x,y),dydt=G(x,y) there is at most one trajectory passing through a given point (x0, y0). Hint: Let C0 be the trajectory generated by the solution x = ϕ0(t), y = ψ0(t), with ϕ0(t0) = x0, ψ0(t0) = y0, and let C1 be the trajectory generated by the solution x = ϕ1(t), y = ψ1(t), with ϕ1(t1) = x0, ψ1(t1) = y0. Use the fact that the system is autonomous, and also the existence and uniqueness theorem, to show that C0 and C1 are the same.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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22.Prove that for the system
dxdt=F(x,y),dydt=G(x,y)
there is at most one trajectory passing through a given point (x0, y0). Hint: Let C0 be the trajectory generated by the solution x = ϕ0(t), y = ψ0(t), with ϕ0(t0) = x0, ψ0(t0) = y0, and let C1 be the trajectory generated by the solution x = ϕ1(t), y = ψ1(t), with ϕ1(t1) = x0, ψ1(t1) = y0. Use the fact that the system is autonomous, and also the existence and uniqueness theorem, to show that C0 and C1 are the same.
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