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ORDINARY DIFFERENTIAL EQUATION Math 2306 Ordinary Diff. Eqns. Name (print): Spring 2020 Test 3 Upload the completed test to D2L by 11:59pm ET, 04/

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ORDINARY DIFFERENTIAL EQUATION

Math 2306 Ordinary Diff. Eqns. Name (print):
Spring 2020
Test 3

Upload the completed test to D2L by 11:59pm ET, 04/20/2020

Sections: 3.2, 3.3, 3.4, 3.5, 3.6

This test is open book and open notes, and it contains 8 pages (including this cover page) and 4
problems.

You are required to show your work on each problem or reason your answer on this exam.

The following rules apply:

Organize your work, in a reasonably neat and
coherent way, in the space provided. Work scat-
tered all over the page without a clear ordering
will receive very little credit.

Mysterious or unsupported answers will not
receive full credit. A correct answer, unsup-
ported by calculations, explanation, or algebraic
work will receive no credit; an incorrect answer
supported by substantially correct calculations and
explanations might still receive partial credit.

Do not write in the table to the right.

Problem Points Score

1 11

2 10

3 14

4 5

Total: 40

Math 2306 Ordinary Diff. Eqns. Test 3 – Page 2 of 8

1. (11 points) Consider the linear system

dx

dt
= 2x + 3y

dy

dt
= 2x + y

(a) (4 points) Compute the eigenvalues and eigenvectors of the matrix for the system.

(b) (2 points) For each eigenvalue, determine a corresponding straight-line solution.

Math 2306 Ordinary Diff. Eqns. Test 3 – Page 3 of 8

(c) (2 points) Sketch manually the phase portrait for the linear system by drawing the
straight-line solutions, equilibrium point(s) as well as other significantly different solu-
tion curves. Do NOT use DE Tools or any other technology for this part or your work will
receive no credit.

(d) (1 point) Identify the equilibrium point for the system as a spiral source, a spiral sink
or a saddle. Reason your answer.

(e) (2 points) Find the general solution.

Math 2306 Ordinary Diff. Eqns. Test 3 – Page 4 of 8

2. (10 points) Consider the linear system

d

Y

dt
=

(
2 8
1 2

)

Y , where

Y (t) =

(
x(t)
y(t)

)
.

(a) (2 points) Compute the eigenvalues of the matrix for the system.

(b) (2 points) For one of the eigenvalues in part (a), compute an eigenvector.

Math 2306 Ordinary Diff. Eqns. Test 3 – Page 5 of 8

(c) (2 points) Sketch manually the phase portrait for the linear system by drawing the

straight-line solutions corresponding to corresponding to

Y1(t) and

Y2(t), equilibrium

point(s) as well as other significantly different solution curves. Do NOT use DE Tools
or any other technology for this part or your work will receive no credit.

(d) (2 points) Determine two solutions

Y1(t) and

Y2(t) for which

Y1(0) and

Y2(0) are linearly

independent.

(e) (2 points) Determine the solution of the linear system subject to the initial condition

Y (0) =

(
2
1

)
.

Math 2306 Ordinary Diff. Eqns. Test 3 – Page 6 of 8

3. (14 points) Consider the initial-value problem

d

Y

dt
=

(
2 4
1 6

)

Y ,

Y (0) =

(
1

6

)
.

(a) (2 points) Compute the eigenvalue(s) of the matrix for the system.

(b) (2 points) Compute an eigenvector corresponding to the eigenvalue found in part (a).

(c) (2 points) Sketch manually the phase portrait, including the solution curve with the

initial condition

Y (0) =

(
1

6

)
and equilibrium solution (s). Do NOT use DE Tools or

any other technology for this part or your work will receive no credit.

(d) (2 points) Sketch roughly x(t) and y(t) graphs of the solution by using the phase
portrait in the part (c).

Math 2306 Ordinary Diff. Eqns. Test 3 – Page 7 of 8

(e) (2 points) Find the general solution.

(f) (2 points) Find the particular solution for the initial condition

Y (0) =

(
1

6

)
.

(g) (2 points) Sketch x(t) and y(t) graphs of the solution found in part (g). You may
use any technology of your choice and reproduce the graph here.

Math 2306 Ordinary Diff. Eqns. Test 3 – Page 8 of 8

4. (5 points) Find the solution to the initial-value problem

y + 2y 3y = 0, y(0) = 6, y(0) = 2.

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