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Practice Test 4 on Differential Equations | MATH 3107, Exams of Differential Equations

Material Type: Exam; Class: Differential Equations; Subject: Mathematics; University: Columbus State University; Term: Spring 2000;

Typology: Exams

Pre 2010

Uploaded on 08/04/2009

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Test 4,Math 3107 Name: _________________________
Spring 2000, Dr. Howard
Instructions: This take-home test is due no later than the beginning of class on Monday,
April 24. You must
work alone
on these problems. You may use your own text, your own
notes, your own calculator, and Maple, but you may use no other aids without my specific
permission. Please show all work and justify all answers.Staple your work with this sh
e
1.Consider the system of differential equations
Y
๎˜
๎˜
09
๎˜
16 Y
a.Find the general solution Y
๎˜
t
๎˜‚
.
b.Sketch the configuration of solutions in the phase plane, indicating any
straight-line solutions.
c.Y
๎˜
t
๎˜‚
๎˜
๎˜
0,0
๎˜‚
is an equilibrium solution of this system.
i.Identify it as a node, a saddle point, a center, or a spiral point, and
ii.determine if it is unstable, stable, or asymptotically stable.
2.Consider the system of differential equations
Y
๎˜
๎˜
715
๎˜
6
๎˜
11 Y
a.Find the general solution Y
๎˜
t
๎˜‚
.
b.Sketch the configuration of solutions in the phase plane, indicating any
straight-line solutions.
c.Y
๎˜
t
๎˜‚
๎˜
๎˜
0,0
๎˜‚
is an equilibrium solution of this system.
i.Identify it as a node, a saddle point, a center, or a spiral point, and
ii.determine if it is unstable, stable, or asymptotically stable.
d.Use Maple to draw the time series solutions, phase plane graph, and space curve
corresponding to the initial condition Y
๎˜
0
๎˜‚
๎˜
๎˜
1,1
๎˜‚
.
3.Compute eAt if Ais the matrix A
๎˜
๎˜
2
๎˜
5
47 .
4.Compute the Laplace transform of the function f
๎˜
t
๎˜‚
๎˜
e
๎˜
tif 0
๎˜‚
t
๎˜‚
5
0if t
๎˜‚
5.
1
pf2

Partial preview of the text

Download Practice Test 4 on Differential Equations | MATH 3107 and more Exams Differential Equations in PDF only on Docsity!

Test 4 , Math 3107 Name: _________________________

Spring 2000, Dr. Howard

Instructions : This take-home test is due no later than the beginning of class on Monday, April 24. You must work alone on these problems. You may use your own text, your own notes, your own calculator, and Maple, but you may use no other aids without my specific permission. Please show all work and justify all answers. Staple your work with this she

1. Consider the system of differential equations

Y ^ 

Y

a. Find the general solution Y  t . b. Sketch the configuration of solutions in the phase plane, indicating any straight-line solutions. c. Y  t   0, 0 is an equilibrium solution of this system. i. Identify it as a node, a saddle point, a center, or a spiral point, and ii. determine if it is unstable, stable, or asymptotically stable.

2. Consider the system of differential equations

Y ^ 

Y

a. Find the general solution Y  t . b. Sketch the configuration of solutions in the phase plane, indicating any straight-line solutions. c. Y  t   0, 0 is an equilibrium solution of this system. i. Identify it as a node, a saddle point, a center, or a spiral point, and ii. determine if it is unstable, stable, or asymptotically stable. d. Use Maple to draw the time series solutions, phase plane graph, and space curve corresponding to the initial condition Y  0   1, 1.

3. Compute e At^ if A is the matrix A 

4. Compute the Laplace transform of the function f  t  

e ^ t^ if 0  t  5 0 if t  5

Test 4 , Math 3107 Name: _________________________

Spring 2000, Dr. Howard

5. Express the function graphed below as a sum of shifted Heaviside functions, i.e. f  t   k 1 H  t  a 1   k 2 H  t  a 2   .

0

1

2

3

1 2 3 4 5 6 7 8

6. In my Maple lecture notes found on the web site, you will find an example demonstrating the solution of a differential equation by Laplace transforms in Maple. Modify that example to solve the following initial value problem: x ^  9 x  1 , x  0   x  0   0.