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Sec. 7.5: Homogeneous Linear Systems with ConstantCoefficients
MATH 351
California State University, Northridge
April 20, 2014
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 1 / 27
Homogeneous linear systems with constant coefficients
x′ = Ax (1)
where A is a constant n × n matrix.
We assume that all the elements of A are real.
If n = 1, then the system reduces to
dx
dt= ax , (2)
its solutions is
x = 0 is the only equilibrium solution if a 6= 0.
If a < 0, other solutions approach x = 0 as t increases, and in this case we saythat x = 0 is ;
If a > 0, other solutions depart from x = 0 as t increases, and in this case we saythat x = 0 is .
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 2 / 27
For systems of n equations, (n ≥ 2)
x′ = Ax (3)
If n ≥ 2,How to get equilibrium solutions?
Questions: Whether other solutions approach this equilibrium solution or departfrom it as t increases; in other words, is x = 0 asymptotically stable or unstable?Or are there still other possibilities?
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 3 / 27
2-dim homogeneous linear system with constant coefficients
If n = 2,
x′ = Ax, i.e.,
(x1x2
)′=
(a bc d
)(x1x2
)(4)
phase plane x1x2-plane includes a direction field of tangent vectors to solutions ofthe system of DEs.
phase portrait a plot includes a representative sample of trajectories for thesystem of DEs.
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 4 / 27
Example 1
Find the general solution of the system
x′ =
(3 00 −2
)x (5)
Solutions:
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 5 / 27
What we learn from Example 1?
? exponential solutions
To solve the general system ofx′ = Ax, (6)
let us try to seek solutions of the form
x = ξert (7)
where the exponent r and the vector ξ are to be determined.
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 6 / 27
Example 2
Consider the system
x′ =
(1 14 −2
)x (8)
Plot a direction field and determine the qualitative behavior of solutions. Then findthe general solution and draw a phase portrait showing several trajectories.
Solutions:
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 7 / 27
Direction Field for the System (8) in Example 2
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x1
x 2
Figure: Direction Field for the system (8) in Example 2
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 8 / 27
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x1
x 2
Figure: Direction Field for the system (8) in Example 2.
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 9 / 27
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x1
x 2
Figure: A phase portrait for the system (8) in Example 2
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 10 / 27
0 0.5 1 1.5 2 2.5 3−25
−20
−15
−10
−5
0
5
10
15
20
25
t
x 1
Figure: Typical solutions of x1 versus t for the system (8) in Example 2
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 11 / 27
Example 3
Consider the system
x′ =
(−2 11 −2
)x (9)
Draw a direction field for this system and find its general solution. Then plot a phaseportrait showing several typical trajectories in the phase plane.
Solutions:
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 12 / 27
Direction Field for the System (9) in Example 3
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x1
x 2
Figure: Direction Field for the system (9) in Example 3
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 13 / 27
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x1
x 2
Figure: Direction Field for the system (9) in Example 3.
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 14 / 27
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x1
x 2
Figure: A phase portrait for the system (9) in Example 3.
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 15 / 27
0 0.5 1 1.5 2 2.5 3−25
−20
−15
−10
−5
0
5
10
15
20
25
t
x 1
Figure: Typical solutions of x1 versus t for the system (9) in Example 3.
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 16 / 27
Example 4
Consider the system
x′ =
(5 −13 1
)x (10)
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 17 / 27
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x1
x 2
Figure: Direction field for the system (10) in Example 4.
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 18 / 27
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x1
x 2
Figure: A phase portrait for the system (10) in Example 4.
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 19 / 27
Example 5
Consider the system
x′ =
(1 11 1
)x (11)
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 20 / 27
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x1
x 2
Figure: Direction field for the system (11) in Example 5.
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 21 / 27
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x1
x 2
Figure: A phase portrait for the system (11) in Example 5.
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 22 / 27
For the general system x′ = Ax,
To solve it, we need find the eigenvalues and eigenvectors by solving the nth degreepolynomial equation
det(A− r I) = 0, (12)
If we assume that A is a real-valued matrix, then we have the following possibilities forthe eigenvalues of A:
All eigenvalues are real and different from each other;
Some eigenvalues occur in complex conjugate pairs;
Some eigenvalues, either real or complex, are repeated.
If the n eigenvalues are all real and different,
eigenvalue ri
eigenvector ξ(i) (the n eigenvectors ξ(1), . . . , ξ(n) are linearly independent)
The corresponding solutions of the system are
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 23 / 27
The general solutions for x′ = Ax
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 24 / 27
If A is real and symmetric (a special case of Hermitian matrices), then
all the eigenvalues r1, . . . , rn must be real;
even if some of the eigenvalues are repeated, there is always a full set of neigenvectors ξ(1), . . . , ξ(n) that are linearly independent (in fact, orthogonal)
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 25 / 27
Example 5
Find the general solution of
x′ =
3 2 42 0 24 2 3
x (13)
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 26 / 27
Summary: x′ = Ax
If A is real-valued
1 All eigenvalues are real and different from each other; (Sec. 7.5)
2 Some eigenvalues occur in complex conjugate pairs; (Sec. 7.6)
3 Some eigenvalues, either real or complex, are repeated. (Sec.7.8)
If A is complex-valued
complex eigenvalues need not occur in conjugate pairs
the eigenvectors are normally complex-valued even though the associatedeigenvalue may be real
the solutions of the system (in general complex-valued) are
MATH 351 (Differential Equations) Sec. 7.5 April 20, 2014 27 / 27