The trajectory of an object moving in the xy-plane is governed by the system of first order ordinary differential equations dx dy = x – 2y dt = -Y, dt with general solution x(t) y(t) ) -« (1). = aj et + a2 (a) Sketch a phase portrait of the system near the critical point at the origin. You should include the following in your sketch, and explain your reasoning to justify your conclusions: • any straight line orbits and their directions; • at least 4 other orbits and their directions, showing the asymptotic behaviour as t → 0 andt → -0o; • the slopes at which the orbits cross the x and y axes; • the velocity vector at (1,1). (b) What is the type and stability of the critical point for this system? (c) Consider the initial conditions x(0) = 0 and y(0) = -1: • solve the system using Laplace transforms; • determine the object's location at time t = 10.
The trajectory of an object moving in the xy-plane is governed by the system of first order ordinary differential equations dx dy = x – 2y dt = -Y, dt with general solution x(t) y(t) ) -« (1). = aj et + a2 (a) Sketch a phase portrait of the system near the critical point at the origin. You should include the following in your sketch, and explain your reasoning to justify your conclusions: • any straight line orbits and their directions; • at least 4 other orbits and their directions, showing the asymptotic behaviour as t → 0 andt → -0o; • the slopes at which the orbits cross the x and y axes; • the velocity vector at (1,1). (b) What is the type and stability of the critical point for this system? (c) Consider the initial conditions x(0) = 0 and y(0) = -1: • solve the system using Laplace transforms; • determine the object's location at time t = 10.
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:The trajectory of an object moving in the xy-plane is governed by the system of first order
ordinary differential equations
dx
= -Y,
dt
dy
= x – 2y
dt
with general solution
æ(t)
y(t)
= aj
e-t + a2
(a) Sketch a phase portrait of the system near the critical point at the origin.
You should include the following in your sketch, and explain your reasoning to justify your
conclusions:
• any straight line orbits and their directions;
• at least 4 other orbits and their directions, showing the asymptotic behaviour as
t → 0 andt → -00;
• the slopes at which the orbits cross the x and y axes;
• the velocity vector at (1, 1).
(b) What is the type and stability of the critical point for this system?
(c) Consider the initial conditions x(0) = 0 and y(0) = -1:
• solve the system using Laplace transforms;
• determine the object's location at time t = 10.
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