Chapter 7
(ii) We now assume that the formula is valid for n = k and show that it is valid then for n = k + 1. Indeed, since
t
t
k+1
= (k + 1)
0
k d ,
applying (5), we conclude that L tk+1 (s) = L (k + 1)
t
k d
0
(s)
1 1 k! (k + 1)! = (k + 1) L tk (s
MA 221 Homework Solutions
Due date: April 1, 2014
8.3 p. 445 17, 19, 21,25 27
17.) Find at least the first four nonzero terms in a power series expansion about x 0 for a
general solution to a given differential equation.
w x 2 w w 0
wx a 0 a 1 x a 2 x 2
MA 221 Homework Solutions
Due date: March 20, 2014
7.5 p. 382 #1, 3, 5, 6, 7, 15, 17, 19
(Underlined problems are to be handed in)
For problems 1, 3, 5, and 6, solve the given initial value problem using the method of
Laplace transforms.
1.)
y 2y 5y 0;
y0
MA 221 Homework Solutions
Due date: March 18, 2014
7.4 pg. 374 - 375 # 1, 3, 5, 7, 9, 12, 15, 17, 21, 23
(Underlined problems are to be handed in)
For problems 1,3, 5, 7, and 9, determine the inverse Laplace transform of the given function.
6
1.) s1 4
MA 221 Homework Solutions
Due date: March 25, 2014
Section 8.2 pg. 434 # 1, 2, 5, 6
(Underlined problems are to be handed in)
1.)
n
n2 1 x 1 n
n0
n1
lim aa n lim
n
n
2 n1 /n 2
1 L
2
2 n /n 1
p 1 2
L
The endpoints of the interval of convergence are
x1
MA 221 Homework Solutions
Due date: March 4, 2014
7.2 pg. 360 # 1, 7, 9, 10, 15, 16, 17, 19
(Underlined problems are to be handed in)
In problems 1, 7 and 9, use Definition 1 to determine the Laplace transform of the given
function.
1.) t
Lts
td
N
lim
Ma 221 - Exam II review
Second Order Differential Equations
Form of general solution
yh c1y1 c2y2
where y 1 and y 2 are linearly independent solutions of the homogeneous equation and
y yh yp
where y p is a [particular] so;lution of the non-homogeneous equ
Ma 221 Homework Solutions Due Date:
January 21, 2014
2.2 pg. 43 # 2, 3, 6, 11, 15, 17 19, 21, 23; (Underlined problems are
handed in)
Page 43
In problems 1, 4 and 5, determine whether the given differential equation is separable.
2)
dy
dx
dy
4y 2 3y1
4y
Ma 221 Homework Solutions Spring 2014
Due January 16, 2014
1.2 p.13-14 #2, 4, 5, 6, 7, 8, 10, 11, 17, 20b, 21b, 22a,b
2.
dy
(a) Show that x x 2 is an explicit solution to x dx 2y on the interval , .
Differentiating x gives:
x 2x
Substituting and for y an
MA 221 Homework Solutions
Due date: January 28, 2014
pg. 61 - 62 Sec. 2.4 #10, 11, 13, 15, 17, 19, 23, 24, 25, 27a, 29
(Underlined Problems are to be turned in.)
In problems 9, 11, 13, 15, 17 and 19, determine whether the equation is exact. If it is, then
Ma 221 Homework Solutions
Due date: January 30, 2014
2.6 p.74 #21, 23, 28;
4.2 p. 165 - 166 #2, 4, 5, 8, 10, 17, 26, 27 29
(Underlined Problems are to handed in.)
2.6 p.74 # 21, 23, 28
For 21, 23 and 28 use the method discussed under "Bernoulli Equations"
MA 221 Homework Solutions
Due February 6, 2014
4.3, pg. 173 # 2, 4, 6, 8, 17, 27, 29b
(Underlined problems are to be handed in)
In problems 2, 4, 6 and 8, the auxilliary equation for the given differential equation has
complex roots. Find a general solu
MA 221 Homework Solutions
Due date: February 13, 2014
4.4 pg. 182 # 6, 14, 12, 13, 15, 17, 21, 23
(Underlined Problems are to be handed in)
In problem 5 determine whether the method or not the method of undetermined coefficients
can be applied to find a
MA 221 Homework Solutions
Due date: March 27, 2014
8.3 p. 445 #1, 3, 5, 7, 11, 12, 15,
(Underlined problems are to be handed in)
In problems 1, 3, 5 and 7 Determine all the singular points of the given differential
equations.
1.)
x 1y x 2 y 3y 0
Divid
MA 221 Homework Solutions
Due date: March 6, 2014
7.3 p. 365 - 366 #1, 3, 11, 21, 25a,b, 31, 33;
turned
(Underlined problems are to be in)
For problems 1, 3 and 11, determine the Laplace transform of the given function using
Table 7.1 and the propert
Ma 221
Chapter 1 - Basic Concepts
Classification of Differential Equations
A differential equation is an equation involving an unknown function and one or more of its
derivatives. Thus it is a relation of the form
F x, y,
dn y
dy
,.,
0
dx
dx n
F is given,
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Ma 221
08S
Exam IB Solutions
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Exam IA Solutions
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shown to obtain full credit
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Ma 221
09S
Exam IB
Solutions
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Solutions - Exam IA
Ma 221
Exam IB Solutions
10S
Solve:
1 25 pts.
2 y 2 2x 1 y
dy
0,
dx
y2 1.
Solution: We write the equation as
2 y 2 dx 2x 1 ydy 0
The M 2 y 2 and N 2x 1 y and
M y 2y N t
Hence the equation is exact and there exists a function ft, y suc
Solutions - Exam IA
Ma 221
Exam IA Solutions
10S
Solve:
1 25 pts.
1 y 2 2t 1 y
dy
0, y0 1
dt
Solution: We write the equation as
1 y 2 dt 2t 1 ydy 0
The M 1 y 2 and N 2t 1 y and
M y 2y N t
Hence the equation is exact and there exists a function ft, y such
Name:_
Lecturer _
Lecture Section: _
Ma 221
07S
Exam IA Solutions
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You may not use a calculator, cell phone, or computer while taking this exam. All work must be
shown to obtain full credit
Ma 221
Laplace Transforms - Chapter 7
Review of Improper Integrals.
Since the Laplace Transform is an improper integral we recall:
Definition. If fx is defined on a x , then we define fxdx by
a
a fxdx lim a fxdx
R
R
provided this limit exists. If the lim
Ma 221
CHAPTER 4 - Linear Differential Equations
We shall now begin a detailed study of the second-order linear differential equation
axy bxy cxy fx
Fundamental theory of second-order linear equations
The following theorem gives information concerning the
Ma 221
Chapter 2 - Special Methods for First Order Equations
Consider the equation
Mx, y Nx, yy 0 1
This equation is first order and first degree. The functions Mx, y and Nx, y are given.
Often we write this as
Mx, ydx Nx, ydy 0 2
Separation of Variables
Ma 221
Series Solutions of Differential Equations
Solution by Power Series
We shall now study ways of solving the second order differential equation
d2y
dy
a 0 xy fx
a 2 x 2 a 1 x
dx
dx
This equation has variable coefficients. In any interval where a 2 x
Ma 221
BOUNDARY VALUE PROBLEMS
Homogeneous Boundary Value Problems
Consider the following problem:
D.E. Ly a 0 x y a 1 x y a 2 x y 0
B.C. 1 ya 1 y a 0
B.C. 2 yb 2 y b 0
axb
2 2 0
1
1
2
2
2
2
1
0
Here 1 , 2 , 1 , and 2 are constants.
Example
y 0
y 0 y 1 0
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ID#:_
Lecture Section: _
Ma 221
05F
Exam IA Solutions
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System._
You may not use a calculator, cell phone, or computer while taking this exam. All work must be
shown to obtain full
Ma 221
Review of Power Series
Please note that there is material on power series at Visual Calculus. Some of this material was used as
part of the presentation of the topics that follow.
The Ratio Test
Recall
Definition: A series a n is called absolutely