Spring 2013
MA 266
Study Guide - Exam # 2
(1)
First Order Dierential Equations. (Separable, 1st Order Linear, Homogeneous, Exact)
(2)
Second Order Linear Homogeneous with Equations Constant Coecients
QUIZ 3 (IN 10 MINUTES.)
(1) A tank originally contains 200 gal of water with 100 lb of salt
in solution. Water containing salt 1 lb/gal is entering at a rate
of 1 gal/min, and the mixture ows out of t
QUIZ 5 (IN 10 MINUTES.)
(1) If we know the solutions of y 3y + 2y = 0 is of type et ,
what are the valid ? (5 points)
(2) Find the general solution of the dierential equation y +y +y =
0. (5 points)
S
QUIZ 6 (IN 10 MINUTES.)
(1) Consider the inhomogeneous dierential equation y 2y
15y = e5t . Knowing that A te5t is a special solution, work out
the coecient A. (5 points)
(2) Find the general solutio
QUIZ 7 (IN 12 MINUTES.)
(1) Consider the equation
u + u = cos t
(1)
subject to the initial condition u(0) = 0 and u (0) = 0
(a) What is the general solution of the homogeneous part, i.e.
u + u = 0? (3
QUIZ 8 (IN 10 MINUTES.)
(1) Consider the equation y y = et + 1.
(a) Does the characteristic polynomial Z () have a repeated
root? (1 point)
(b) Find the general solution. (4 points)
(2) Knowing that y
Midterm Exam 1, math 266, Fall 2009
Print your last name:
Circle the time of your class:
Instructions:
9:30am,
rst name:
10:30am.
1. This exam contains 10 pages. The last page is left intentionally bl
Midterm Exam 1, math 266, Fall 2010, version 1
Print your last name:
rst name:
Please circle your answers
1. A B C
D
E
2. A B C
D
E
3. A B C
D
E
5. A B C
D
E
6. A B C
D
E
7. A B C
D
4. A B C
D
E
E
Gra
Midterm Exam 1, math 266
Feb. 19th, Spring 2009
rst name:
Print your last name:
Circle the time of your class:
8:30am,
9:30am,
10:30am,
12:30pm.
Instructions:
1. This exam contains 10 pages, including
QUIZ 2 (IN 10 MINUTES.)
dy
x2
y
(1) Solve the equation
=2
+ . (10 points)
2
dx
x +y
x
Solutions.
y
(1) Observing what on the R.H.S is a function of
we use the
x
y
substitution v = :
x
dv
1
The equatio
QUIZ 1 (IN 15 MINUTES.)
dy
= 3y . (4 points)
(1) Find the general solution of
dt
dy
(2) Solve the equation (2+ t2 ) +2ty = 2t subject to the condition
dt
y (0) = 2. (6 points)
Solutions.
1
dy = 3dt, w
Consider the IVP
dy = t(1 - y 2 ), dt (t) = tanh
y (0) = 0.
1 2 2t
The exact solution is .
Direction field and solution:
1.5 1.0 0.5 0.0 0.5 1.0 1.5 0 1 2 3 4
1 / 17
dy = t(1 - y 2 ), y (0) = 0. dt We
f (t) = L1 cfw_F (s)
F (s) = Lcfw_f (t)
1.
1
1
s
2.
eat
1
sa
3.
tn
n!
sn+1
4.
tp (p > 1)
5.
sin at
6.
cos at
7.
sinh at
8.
cosh at
s
s2 a2
9.
eat sin bt
b
(s a)2 + b2
10.
eat cos bt
sa
(s a)2 + b2
11.
Spring 2013
MA 266
Study Guide - Exam # 2 & FINAL EXAM
(1)
First Order Dierential Equations. (Separable, 1st Order Linear, Homogeneous, Exact)
(2)
Second Order Linear Homogeneous with Equations Consta
Spring 2013
MA 266
Study Guide - Exam # 1
(1) Special Types of First Order Equations
dy
+ p(t)y = g(t)
dt
I First Order Linear Equation (FOL):
Solution Method :
y=
1
(t)
[
]
(t)g(t) dt + C , where (t)
. ,7 Spring 2013
.MA 266 Exam # 1
Name _ PUID#
(5 pts) 1. The largest open interval for which the solution of the following rst order linear equation with
initial condition is certain to exist is :
REVIEW SHEET 1
(1) Determine the values of r such that y = tr is a solution of the equation
t2 y + 4ty + 2y = 0.
(2) Let y be a solution of the initial value problem
1
1
y = x, x > 0, and y (1) = ,
y
REVIEW SHEET 2
(1) Consider the homogeneous dierential equation y 4y + y 4y = 0.
(a) Find the characteristic equation and explain the meaning of the equation?
(b) Find the fundamental solutions y1 , y
QUIZ 9 (IN 12 MINUTES.)
a
(1) (Multiple choice.) Given L[sin at] = 2
and L[cos at] =
s + a2
s
, then what is
2 + a2
s
L[sin2 at + cos2 at]?
(4 points)
a
s
A. 2
+2
2
s +a
s + a2
B. 1
2
2
s
a
+2
C.
2 +
Midterm Exam 1, math 266, Spring 2010, version 1
Print your last name:
rst name:
Please circle your answers
1. A B C
D
E
2. A B C
D
E
3. A B C
D
E
5. A B C
D
E
6. A B C
D
E
7. A B C
D
E
Grade
1. (10 p
Midterm Exam 1, math 266, Fall 2013
Print your last name:
Circle the time of your class:
Instructions:
(061) 2:30pm,
rst name:
(031) 3:30pm.
1. This exam contains 9 pages. The last page is left intent
COMBINE TRIGONOMETRIC FUNCTIONS SOP
When we solve the dierential equation
ay + by + cy = 0,
we will have the general solution
(1)
4ac b2
4ac b2
c1 cos
y=e
t + c2 sin
t
2a
2a
if b2 < 4ac. This notes is
MA26600
FINAL EXAM INSTRUCTIONS
NAME
December 13, 2010
INSTRUCTOR
1. You must use a #2 pencil on the marksense sheet (answer sheet).
2. On the mark-sense sheet, ll in the instructors name (if you do n
MA 26600
FINAL EXAM INSTRUCTIONS
Dec 14, 2009
NAME
INSTRUCTOR
1. You must use a #2 pencil on the marksense sheet (answer sheet).
2. If the cover of your question booklet is GREEN, write 01 in the TEST
Boyce/10ed/Ch4
Chapter Review Sheets for
Elementary Differential Equations and Boundary Value Problems, 10e
Chapter 4: Higher Order Linear Equations
Definitions:
nth Order Linear ODE
Fundamental Set
Boyce/10ed/Ch3
Chapter Review Sheets for
Elementary Differential Equations and Boundary Value Problems, 10e
Chapter 3: Second Order Linear Equations
Definitions:
Linear and nonlinear
Homogeneous, No
MA 266
FINAL EXAM INSTRUCTIONS
May 8, 2010
NAME
INSTRUCTOR
1. You must use a #2 pencil on the marksense sheet (answer sheet).
2. On the mark-sense sheet, ll in the instructors name (if you do not know
MA 26600
FINAL EXAM INSTRUCTIONS
May 6, 2009
NAME
INSTRUCTOR
1. You must use a #2 pencil on the marksense sheet (answer sheet).
2. If the cover of your question booklet is GREEN, write 01 in the TEST/
MA26600
FINAL EXAM INSTRUCTIONS
NAME
December 13, 2010
INSTRUCTOR
1. You must use a #2 pencil on the marksense sheet (answer sheet).
2. On the mark-sense sheet, ll in the instructors name (if you do n