Alyssa Kritz Milliken MAT 275 Fall 2019
Assignment Section 7.5 Homogeneous systems Constant Coefficients due 12/07/2019 at 11:59pm MST
1. (1 point) Suppose that the matrix A has the following
eigenvalues and eigenvectors:
λ1=1 with ~v1=3
1.
and
λ2=−1 with ~v2=0
1.
Write the solution to the linear system~r0=A~rin the follow-
ing forms.
A. In eigenvalue/eigenvector form:
x(t)
y(t)=c1 et+c2 et
B. As two equations: (write ”c1” and ”c2” for c1and c2)
x(t) =
y(t) =
Note: if you are feeling adventurous you could use other
eigenvectors like 4~v1or −3~v2.
2. (1 point)
Solve the IVP
dx
dt =8−4
4−2x,x(0) = −3
0
x(t) = .
3. (1 point) Consider the system of differential equations
dx
dt =−1.2x+y,
dy
dt =1.25x−3.2y.
For this system, the smaller eigenvalue is and
the larger eigenvalue is .
Use the phase plotter pplane9.m in MATLAB to determine
how the solution curves behave.
•A. The solution curves race towards zero and then veer
away towards infinity. (Saddle)
•B. All of the solution curves run away from 0. (Unsta-
ble node)
•C. The solution curves converge to different points.
•D. All of the solution curves converge towards 0. (Sta-
ble node)
The solution to the above differential equation with initial
values x(0) = 6,y(0) = 6 is
x(t) = ,
y(t) = .
4. (1 point)
Consider the systems of differential equations
dx
dt =0.4x+0.5y,
dy
dt =1.5x−0.6y.
For this system, the smaller eigenvalue is and
the larger eigenvalue is .
Use the phase plotter pplane9.m in MATLAB to determine
how the solution curves behave.
•A. The solution curves race towards zero and then veer
away towards infinity. (Saddle)
•B. The solution curves converge to different points.
•C. All of the solution curves converge towards 0. (Sta-
ble node)
•D. All of the solution curves run away from 0. (Unsta-
ble node)
The solution to the above differential equation with initial
values x(0) = 2,y(0) = 8 is
x(t) = ,
y(t) = .
5. (1 point)
Consider the systems of differential equations
dx
dt =0.3x−0.8y,
dy
dt =−0.2x+0.9y.
For this system, the smaller eigenvalue is and
the larger eigenvalue is .
1
Use the phase plotter pplane9.m
in MATLAB to determine how the solution curves behave.
•A. All of the solution curves run away from 0. (Unsta-
ble node)
•B. The solution curves converge to different points.
•C. The solution curves race towards zero and then veer
away towards infinity. (Saddle)
•D. All of the solution curves converge towards 0. (Sta-
ble node)
The solution to the above differential equation with initial
values x(0) = 2,y(0) = 3 is
x(t) = ,
y(t) = .
2
