Homework Assignment 11 in Differential Equations, MATH308-FALL 2015
Total number of points: 150
due November 4, 2015 Sections covered: end of 7.5 (the case when there are repeated eigenvalues
and a basis of eigenvectors)& 7.8 (when there are repeated eigenvalues but no basis of eigenvectors)
1. Given the following system of linear differential equations:
 0
 x1 = x2 − 4x3
x0 = −9x1 + 6x2 − 12x3
 20
x3 = −3x1 + x2 − x3
(a) Find the eigenvalues of the corresponding matrix and for each eigenvalue determine its algebraic and geometric multiplicities;
(b) Find the general solution.
2. Given the following system of linear differential equations:
0
x1 = x1 − 29 x2
x02 = 2x1 + 7x2
(1)
(a) Find the general solution of the system (1).
x1 (t)
(b) If x(t) =
is a solution of (1), what is the limit of x(t) as t → −∞. Does this limit
x2 (t)
depend on initial conditions?
(c) Find the solution of the system (1) satisfying the initial conditions: x1 (0) = 3,
x2 (0) = −4.
3. Given the following system of linear differential equations:
 0
 x1 = 5x + 4x2 − 5x3
x0 = −16x1 − 4x2 + 15x3
 20
x3 = 2x1 + 4x2 − 2x3
(a) Find the eigenvalues of the corresponding matrix and for each eigenvalue determine its algebraic and geometric multiplicities;
(b) Find the general solution.
1