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.
tn eat
n!
(s a)n+1
12.
uc (t)
ecs
s
13.
uc (t)f (t c)
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 the tank at a rate of
2 gal/min.
(a) Find the amount of
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)
Solutions.
(1) Plug et into the equation and we can get
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 solution of the dierential equation 9y 12y +
4y = 2. (5 points
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 points)
(b) Lets use the method of variation parameter
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 y1 = cos t, y2 = sin t, y3 = t cos t, y4 = t sin t, y5 =
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 blank, which you may use as scrap
paper.
2. This exam con
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
Grade
1. (10 pts) Determine the values of r such that y =
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 the cover page. The last page is left intentially blan
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 equation can be rewritten as x
+v =
+ v.
dx
1 + v2
Hence we it
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, which
y
implies ln |y | = 3t + A, where A is a constant.
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 now apply Euler's method with h = 0.2. Note that f (t,
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 Constant Coecients .
The dierential equation ay + by + cy = 0
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 .
The dierential equation ay + by + cy = 0 has Characte
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) = e
p(t) dt
dy
= h(x) g(y)
dx
II Separable Equation (S
. ,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 :
{t(t+1)y+100(t2)y= (til3; A. 4<t<3
1W ; 0<t<3
we):
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
2x
3
then what is y (4)?
1
2
REVIEW SHEET 1
(3) Which e
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 , y2 , y3 .
(c) Is W (y1 , y2 , y3 ) = 0?
(2) If the funda
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 + a2
2
s
s +a
1
D.
s
1
1
a2
E. 2
+2
s s + a2 s + a2
(2) U
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 pts) How many of the following dierential equations are
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 intentionally blank, which you may use as scrap
paper.
2. Thi
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 aimed to combine the trigonometric functions in the
4a
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 not know, write down
the class meeting time and location
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/QUIZ NUMBER
boxes and blacken in the appropriate space
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 of Solutions, General Solution
Homogeneous and Nonhom
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, Nonhomogeneous
Characteristic Equation Wronskian
Genera
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, write down
the class meeting time and location) and t
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/QUIZ NUMBER
boxes and blacken in the appropriate spaces
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 not know, write down
the class meeting time and location