VANDERBILT UNIVERSITY
MATH 198 METHODS OF ORDINARY DIFFERENTIAL EQUATIONS.
PRACTICE MIDTERM II.
The Laplace transform.
The table below indicates the Laplace transform F (s) of the given function f (t).
f (t)
F (s)
1
1
s
1
t
s2
n!
n
t
sn+1
1
eat
sa
n!
n ea
Selected Math 198 Assignment Solutions
These solutions are mostly for the even numbered problems that were assigned for homework. Some of the lengthier calculations are abbreviated, but these almost always just
require basic arithmetic or the evaluation o
VANDERBILT UNIVERSITY
MATH 198 METHODS OF ORDINARY DIFFERENTIAL EQUATIONS
SOLUTIONS TO THE PRACTICE MIDTERM.
Question 1. For each equation below, identify the unknown function, classify the equation as linear
or non-linear, and state its order.
dy
y
+ = 0
VANDERBILT UNIVERSITY
MATH 198 METHODS OF ORDINARY DIFFERENTIAL EQUATIONS
EXAMPLES OF SECTIONS 4.6 AND 4.7.
Question 1. Find the particular solution yp of the equation
x2 y 4xy + 6y = x3 .
(1)
Question 2. Show that the formula given in class for yp , name
VANDERBILT UNIVERSITY
MATH 198 METHODS OF ORDINARY DIFFERENTIAL EQUATIONS.
PRACTICE FINAL SOLUTIONS.
The Laplace transform.
The table below indicates the Laplace transform F (s) of the given function f (t).
f (t)
F (s)
1
1
s
1
t
s2
n!
n
t
sn+1
1
eat
sa
n!
VANDERBILT UNIVERSITY
MATH 198 METHODS OF ORDINARY DIFFERENTIAL EQUATIONS.
PRACTICE MIDTERM II.
The Laplace transform.
The table below indicates the Laplace transform F (s) of the given function f (t).
f (t)
F (s)
1
1
s
1
t
s2
n!
n
t
sn+1
1
eat
sa
n!
n ea
VANDERBILT UNIVERSITY
MATH 198 METHODS OF ORDINARY DIFFERENTIAL EQUATIONS
EXAMPLES OF SECTION 2.3.
Question 1. Find a solution to the initial value problem
(50 + t)x + x 8t = 400,
x(0) = 10,
where t 0.
Question 2. Consider the two interconnected tanks sho
VANDERBILT UNIVERSITY
MATH 198 METHODS OF ORDINARY DIFFERENTIAL EQUATIONS
PRACTICE MIDTERM.
Question 1. For each equation below, identify the unknown function, classify the equation as linear
or non-linear, and state its order.
(a) y
(b) x
dy
y
+ = 0.
dx
VANDERBILT UNIVERSITY
MATH 198 METHODS OF ORDINARY DIFFERENTIAL EQUATIONS.
PRACTICE FINAL.
1
2
MATH 198 - PRACTICE FINAL
The Laplace transform.
The table below indicates the Laplace transform F (s) of the given function f (t).
f (t)
F (s)
1
1
s
1
t
s2
n!
Q, N I
TESTl lag ' }W%L,.
Spring, 2005 ' y/ , I
h4AITllQ8 Hemorrnedge_i_iz;_":i_ii _ _
This is a closed note, closed book test. No calculators are allowed. This test is subject to
the honor code. SHOW ALL WORK! No work, no credit. Good Luck!
[2 each] 1. D
Vanderbilt University
Math 198, Section 2, spring 2007
Test 2
Name (Print): f, if) (,3 {A
l.‘ A, ‘ x 7.“
(First name) (Last name)
Last 4 digits of Student ID: (A. l I ‘
N
Please write and sign the honor pledge below. Calculators are not allowed.
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m
Vanderbilt University
Math 198, Section 2, spring 2007
Test 1
l, ‘ (I 7 p "-1
(First name) (Last name)
Name (Print):
Last 4 digits of Student ID:
Please write and Sign the honor pledge below. Calculators are allowed. (1) Find values of m for which
NAME (PRINT):
Solutions
MATH 198: Methods of Ordinary Dierential Equations
TEST 1, Spring 2014
Problem
Points
Initial Value Problems
20
First Order ODEs
40
Second Order ODEs
40
Total
Your Score
100
Rules of the exam
You have 75 minutes to nish this exam.
NAME (PRINT):
Solutions
MATH 198: Methods of Ordinary Dierential Equations
TEST 2, Spring 2014
Problem
Points
Laplace Transforms
30
Laplace Transforms and IVPs
45
Series Solutions
25
Total
Your Score
100
Rules of the exam
You have 75 minutes to nish this
M AT“ 198: Methods of Ordinary Uiﬂ'orontini Equations
Spring 20! I
Exam 3 (‘Hl‘l‘t‘i‘iiulm
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Vanderbilt University
Math 198, Section 2, spring 2007
Test 3
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Name (Punt): ' x j; w. a
(First name)
Last 4 digits of Student ID: / E;
Please write and sign the honor pledge below. Calculators are not alloWed. (1) Solve the second order equati
VANDERBILT UNIVERSITY
MATH 198 METHODS OF ORDINARY DIFFERENTIAL EQUATIONS
EXAMPLES OF SECTION 2.2.
Question 1. The intensity I of the light at a depth of x meters below the surface of a lake satises
the dierential equations I = 1.4I.
(a) At what depth is
VANDERBILT UNIVERSITY
MATH 198 METHODS OF ORDINARY DIFFERENTIAL EQUATIONS
EXAMPLES OF SECTIONS 7.2 AND 7.3.
Question. Suppose that the Laplace transform Lcfw_f (s) = F (s) exists for s > . Show that
Lcfw_eat f (t) = F (s a),
for s > + a.
Solution. We use
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