MATH 219 - Dierential Equations
Quiz 9
Date
Name
1. Find the given transform or inverse transform.
(a) L
cos(4t)U
L
cos(4t)U
2
t
2
t
= e(/2)s L cfw_cos (4t) =
(b) L 1
L 1
=L
cos 4 t
se(/2)s
s2 + 16
2
U
t
1
(s 1)4
1
s4
ss1
(c) L 1
6
s4
ss1
1
= et t3
6
es
Eulers Method
Name:
General Solution
Find the general solution to y = 2y + 4x using the techniques from sections 2.5 or 4.4.
Use a graphing calculator to plot solutions corresponding to each of the initial conditions.
y(0) = 1, y(0) = 2, y(0) = 3, y(0) =
The Laplace Transform of a Periodic Function
Periodic Functions
Denition. A function f is periodic (with period T ) if f (t + T ) = f (t) for all t in the
domain of f .
A periodic function has regular repetitive behavior.
Example. f (t) = sin(t) has perio
Cauchy-Euler Equations
Name:
1
Cauchy-Euler Equations
The following are second order linear equations with non-constant coecients. Write the
coecients of each equation next to the equation.
1. x2 y + 5xy + 3y = 0
2. x2 y 3xy + 4y = 0
3. 4x2 y + y = 0
4. x
Homogeneous Linear Equations with Constant
Coecients
Name:
1
Distinct real roots
Lets solve a second order equation: y + 4y + 3y = 0. We start with a guess at the form
of a solution based on the rst order case. The number m is a real number that will be
d
Cauchy-Euler Equations
Name:
1
Cauchy-Euler Equations
The following are second order linear equations with non-constant coecients. Write the
coecients of each equation next to the equation.
1. x2 y + 5xy + 3y = 0
x2
5x
3
2. x2 y 3xy + 4y = 0
x2
3x
4
3. 4x
Integrating factors and linear equations
Name:
Integrating factors
1. Verify that the equation
y(x + y + 1)dx + (x + 2y)dy = 0
is not exact.
2. Consider the function (x) = ex . This function is called an integrating factor and it
helps us solve the equati
Using the Laplace transform to solve initial value
problems
Name:
1
Laplace Transform
Fill in the formulas below about the Laplace transform. They will be used in solving initial
value problems in the next sections.
L cfw_tn =
n!
k
k
L cfw_sinh(kt) = 2
2
The Laplace Transform of a Periodic Function
Periodic Functions
Denition. A function f is periodic (with period T ) if f (t + T ) = f (t) for all t in the
domain of f .
A periodic function has regular repetitive behavior.
Example. f (t) = sin(t) has perio
Homogeneous Linear Equations with Constant
Coecients
Name:
1
Solutions
Distinct real roots
Now lets solve a second order equation: y + 4y + 3y = 0. We start with a guess at the
form of a solution based on the rst order case. The number m is a real number
Eulers Method
Name:
1
General Solution
Find the general solution Rto y = 2y + 4x using the techniques from sections 2.5 or 4.4.
Integrating factor:
e 2dx = e2x
e2x (y + 2y) = 4x(e2x )
d 2x
(e y) = 4xe2x
dx
General solution:
e2x y = 2xe2x e2x + c
y = 2x 1
Matrices
Date
Name
Introduction
Denition. A matrix A is a rectangular array of numbers or functions.
a11 a12 . . . a1n
a21 a22 . . . a2n
A=
.
.
.
.
.
.
am1 am2 . . .
amn
If a matrix A has m rows and n columns we say that A is an m n matrix (read m by
The Laplace Transform
Name:
1
Laplace transform and inverse transform
Denition. Let f be a function dened for t 0. Then the integral
L cfw_f (t) =
est f (t)dt
0
is said to be the Laplace transform of f provided the integral converges.
Fill in the followin
The Laplace Transform
Name:
1
Laplace transform and inverse transform
Denition. Let f (t) be a function dened for t 0. Then the integral
L cfw_f (t) =
est f (t)dt
0
is said to be the Laplace transform of f provided the integral converges.
Fill in the foll
Matrices
Date
Name
Introduction
Denition. A matrix A is a rectangular
a11
a21
A=
.
.
.
am1
array of numbers or functions.
a12 . . . a1n
a22 . . . a2n
.
.
.
am2 . . . amn
If a matrix A has m rows and n columns we say that A is an m n matrix (read m by
Integrating factors and linear equations
Name:
Integrating factors
1. Verify that the equation
y(x + y + 1)dx + (x + 2y)dy = 0
is not exact.
M
= x + 2y + 1
y
N
=1
x
Not equal. Not exact.
2. Consider the function (x) = ex . This function is called an integ
MATH 219 - Dierential Equations
Quiz 6
Date
Name
Find the general solution to the dierential equation. Use the method of variation of parameters.
y + y = csc x
1. General solution of y + y = 0.
m2 + 1 = 0
m = i
y1 = cos x, y2 = sin x
yc = c1 cos x + c2 si
MATH 219 - Dierential Equations
Quiz 5
Name
Date
Solutions
1. Use the method of undetermined coecients to nd the general solution.
y 4y + 4y = 1 + 5 sin x
1. Characteristic equation:
m2 4m + 4 = 0
(m 2)2 = 0
General solution to homogeneous equation.
yc =
MATH 219 - Dierential Equations
Quiz 8
Date
Name
Find the Laplace transforms of the functions.
1. f (t) = (1 + e2t )2
L cfw_(1 + e2t )2 = L cfw_1 + 2e2t + e4t =
1
2
1
+
+
s s2 s4
2. f (t) = sin(2t) + cos(3t)
L cfw_sin(2t) + cos(3t) =
s2
s
2
+ 2
+4 s +9
MATH 219 - Dierential Equations
Quiz 6
Name
Date
1. A 1 kg mass is attached to a spring whose constant is 16 N/m and the entire system is submerged in a liquid that imparts a damping force numerically equal to 8
times the instantaneous velocity and the ma
MATH 219 - Dierential Equations
Quiz 4
Date
Name
1. State the type of the equation. It may be of more than one type; you only need to
state one. Then, nd the general solution to the equation.
x2 y + x(x + 2)y = ex
y +
Type:
Linear
x+2
ex
y= 2
x
x
(x) = e
MATH 219 - Dierential Equations
Quiz 1
Date
Name
y x
1. Verify that y = x 2 ln x is a solution to the dierential equation y = + .
x y
If y = x 2 ln x then,
y =
x 2 ln x
x
y x
1
1
=
+
= +
2 ln x + (2 ln x)1/2 (2/x) = 2 ln x +
2
x
x y
2 ln x
x 2 ln x
y x
MATH 219 - Dierential Equations
Exam 2
Date
Name
Read directions and show all work. Answers without support will not be given full credit.
(30 points) 1. Consider the second order dierential equation y + 4y = sec(2x).
(a) Is the equation a Cauchy-Euler eq
MATH 219 - Dierential Equations
Quiz 10
Date
Name
1. Solve the dierential equation subject to the initial conditions.
y 5y + 6y = U (t 1), y(0) = 0, y (0) = 1
es
Side: Partial fractions
s2 Y (s)sy(0)y (0)5(sY (s)y(0)+6Y (s) =
s
1
1
1
es
=
+
2
s Y (s) 1 5s
MATH 219 - Dierential Equations
Exam 3
Date
Name
Read all directions carefully. Show all work. Answers without support will not be given full
credit.
1. Find the given transform or inverse transform.
(15 points) (a) L et sin 3t
L et sin 3t =
3
3
= 2
2+9
(
MATH 219 - Dierential Equations
Quiz 3
Date
Name
State the type of each of the following equations. (Separable, exact or homogeneous) Equations may be of more than one type; you only need to state one. Then, nd the general
solution to each of the equation
MATH 219 - Dierential Equations
Exam 1
Date
Name
Read all directions carefully. Show all work. Answers without support will not be given full
credit.
(Each 15 pts) 1. In parts (a), (b), and (c), state the type of the equation. (Separable, exact, homogeneo
MATH 219 - Dierential Equations
Quiz 2
Name
Date
Solutions
1. Solve the dierential equation.
dy
y sin x
=
dx
1 + 2y 2
1 + 2y 2
dy = sin x dx
y
1 + 2y 2
dy = sin x dx
y
1
+ 2y dy = cos x + C
y
ln |y| + y 2 = cos x + C
ln y + y 2 = cos x + C
2. Solve the di
Using the Laplace transform to solve initial value
problems
Name:
1
Laplace Transform
Fill in the formulas below about the Laplace transform. They will be used in solving initial
value problems in the next sections.
L cfw_tn =
L cfw_sin(kt) =
L cfw_sinh(