A2 For the system da dt = x(x - 2), dy dt = -xy, (a) Find a and b, such that H = xy + ax + by is a first integral for the system. Sketch the level set H= 0. (b) Sketch a phase portrait of the system indicating all fixed points. Are these fixed points isolated? Are they hyperbolic? (c) Find the general solution of the a-equation and use the first integral to find the corre- sponding y(t). Hence find the solution of the initial value problem with initial condition x(0) = xo, y(0) = 3o, where 0 0. What is the limit of (z(t), y(t)) as t→∞o? (d) Find the Hamiltonian system given by H and sketch the phase portrait.
A2 For the system da dt = x(x - 2), dy dt = -xy, (a) Find a and b, such that H = xy + ax + by is a first integral for the system. Sketch the level set H= 0. (b) Sketch a phase portrait of the system indicating all fixed points. Are these fixed points isolated? Are they hyperbolic? (c) Find the general solution of the a-equation and use the first integral to find the corre- sponding y(t). Hence find the solution of the initial value problem with initial condition x(0) = xo, y(0) = 3o, where 0 0. What is the limit of (z(t), y(t)) as t→∞o? (d) Find the Hamiltonian system given by H and sketch the phase portrait.
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:A2 For the system
da
dt
= x(x - 2),
dy
dt
= -xy,
(a) Find a and b, such that H = xy + ax + by is a first integral for the system. Sketch the
level set H= 0.
(b) Sketch a phase portrait of the system indicating all fixed points. Are these fixed points
isolated? Are they hyperbolic?
(c) Find the general solution of the a-equation and use the first integral to find the corre-
sponding y(t). Hence find the solution of the initial value problem with initial condition
x(0) = xo, y(0) = 30, where 0<xo < 2, 3o > 0. What is the limit of (x(t), y(t)) as
t → ∞o?
(d) Find the Hamiltonian system given by H and sketch the phase portrait.
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