Graphing the First Order Differential Equations in Maple
Basic addition, subtraction, multiplication and division: +, -, * and /
Always restart each problem with: >restart:
1. Load the DEtools and plotspackages. You have to do this only once, when you
sta
Test 1 Dierential Equations Solutions 2/15/02
Write your name on the back of your test. Show all work. Where possible test your solutions.
1.(10) Use the guess and test method to solve the second order dierential equation initial value
problem:
d2 y
dy
+
Test 1 Dierential Equations 2/15/02
Write your name on the back of your test. Show all work. Where possible test your solutions.
1.(10) Use the guess and test method to solve the second order dierential equation initial value
problem:
d2 y
dy
+ 6 + 8y = 0
Test 2 Dierential Equations Spring 2002, makeup due 4/22/02
Write your name on the back of your test. Show all work. You must check your work to receive credit.
1.(14) For the dierential equation
d2 y
dy
+ 4 + 13y = sin 3t
2
dt
dt
a) compute the general s
Test 2 Dierential Equations Spring 2003, April 7-8, Solutions
Write your name on the back of your test. Show all work. Where possible test your solutions.
1.(25) For the dierential equation
d2 y
+ 10y = 2 cos 3t
dt2
a) compute the general solution,
Soluti
Quiz 3.4 Make up Solution, 3/26-3/31/03
1) Solve the 2-dimensional system of dierential equations
dY
dt
2
2
Y
13 4
=
1
.
1
Solution: First we nd the eigenvalues of the matrix:
with initial condition Y (0) =
2 T + D = 0
T
= 6
D = (2)(4) 2(13) = 34
2 + 6 +
Homework 1, Selected Problems
1.1 6. Suppose two students memorize lists according to the same model
dL
= 2(1 L).
dt
a) If one of the students knows one-half of the list at time t = 0 and the other knows none of the list,
which student is learning most ra
Practice Material for Final Dierential Equations
Note that this list of questions is not exhaustive. You should also be prepared to answer questions
such as those on test 1, test 2, the make up test for test 2, inclass quizzes, and homework assignments.
A
Practice Material for Final Dierential Equations
Note that this list of questions is not exhaustive. You should also be prepared to answer questions
such as those on test 1, test 2, practice tests, inclass quizzes, and homework assignments. A copy of
p. 6
Quiz 2 Dierential Equations Solutions
For the initial value problem:
dy
dt
=
1
, y (0) = 0
(y + 2)(t 1)
(a) Find a formula for the solution.
Solution: The dierential equation is separable, so we have
(y + 2)dy =
1
dt
t1
y2
+ 2y = ln |t 1| + C
2
y2
+ 2y ln
3.4 (b) Classification of Systems with
Complex Eigenvalues
Ei
Linear Combinations of Sines and Cosines
x(t ) = 3sin(2t ) + 4 cos(2t )
1
Amplitude and Phase
x(t ) = a sin t + b cos t , a > 0
Half
Half-Amplitude : A = a 2 + b 2
b
Phase angle : = arctan
a
x
3.3 Systems with Real Eigenvalues
A Diagonal System
dx
= 3x
3 0 dt
dY
= AY, A =
:
dy
dt
0 2
= 2y
dt
1
1 = 3, V1 =
0
Eigenvalues and eigenvectors :
= 2, V = 0
2
2
1
1
0 k e 3t
General Solution : Y(t ) = k1e 3t + k2 e 2t = 1 2t
0
1 k2 e
Straight
3.1 Properties of Linear Systems
Properties of Linear Systems
Linear Systems
Pauls Rock and Roll CDs & Bobs Opera Only Discs:
x(t ) = daily profit of Paul's store
y (t ) = daily profit of Bob's store
dx
= ax + by
dt
dy
= cx + dy
dt
1
Interpretation of Coe
8/21/2012
3.2 Straight-Line Solutions
Straight
Solutions
The Plan
dx
= ax + by
dt
dy
= cx + dy
dt
a b
dY
= AY, A=
dt
c d
1) Find two linearly independent solutions Y1 (t), Y2 (t)
2) Use the principle of linearity to:
a) Write the general solution : Y(t )
Test 2 Dierential Equations Spring 2002, makeup due 4/22/02
Write your name on the back of your test. Show all work. You must check your work to receive credit.
1.(14) For the dierential equation
dy
d 2y
+ 4 + 13y = sin 3t
2
dt
dt
a) compute the general s