Homework Assignment 13 in Differential Equations, MATH308-FALL 2015
due November 16, 2015
Sections covered:9.3 (locally linear systems), and 7.9 & 3.6 (the method of variation of parameters for
nonhomogeneous systems and dufferential equations of second order)
1. For the following systems
x0
y0
x2 + y 2 − 13
(x − 2)(y − 2)
=
=
(a) Determine all critical points;
(b) Find the corresponding linear system near each critical point;
(c) Based on this linear systems determine the type of each critical point and the stability properties of the corresponding critical points (whether they are stable, asymptotically stable, or
unstable)?
2. Use the method of variation of parameters to find the solution of the following initial value problems:
(a)
x01 = x1 − 2x2 + sin 2t
,
x02 = x1 − x2 − 2 cos 2t
x1 (0) = 3, x2 (0) = −2.
(b)
y 00 − 2y 0 + y =
et
,
1 + t2
1
y(0) = −2, y 0 (0) = 1