Homework Assignment 12 in Differential Equations, MATH308-Fall 2015
due November 9, 2015
Topics covered : The Phase Plane: linear systems (section 9.1): types of critical (equillibrium) points
and stability; sketch of the phase portrait for planar linear systems.
1. For each of the following systems
i) Find and classify the critical (equilibrium) point and determine whether it is stable, asymptotically stable, or unstable;
ii) Sketch the phase portrait of the system (indicating direction of motion along trajectories by
arrows and the direction of eigenvectors, if relevant):
(a)
x01
x02
= −2x1 + 5x2
= −5x1 + 6x2 ,
(b)
x01
x02
= 4x1 + 2x2 − 16
= 6x1 − 7x2 − 4,
x01
x02
= 3x1 − 9x2
= 2x1 − 3x2 − 6,
x01
x02
= 7x1 − 8x2 + 9
= 2x1 − x2 ,
(c)
(d)
(e)
x01
x02
−10x1 + 5x2 + 20
−7x1 + 2x2 + 10.
=
=
1