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
MA 266 Lecture 9
Section 2.5 Autonomous Equations and Population
Dynamics
(4 914 mm M) #3
A differential equation is called L if it has the form \_ V f' L 9 >
00% L
We know how to solve autonomous equations because they are 9% 2 W L iii this sec-
tion, w
MA 266 Lecture 19
Section 3.7 Mechanic and Electrical Vibrations
In this section we use second order linear equations to model some physical processes.
Motion of a Mass on a Spring
Consider a mass m hanging at rest on the end of a vertical spring of origi
MA 266 Lecture 28
6.5 Impulse Functions
In this section, we consider the differential equation with an impulsive nature, i.e.,
ay + by + Cy = W),
Where g(t) is large during a short interval to - 7' < t < to + T for some 7' > 0, and is otherwise zero.
The
MA 266 Lecture 10
Section 2.6 Exact Equations and Integrating Factors
In the section, we consider a special class of rst order equations known as exact equations.
Example 1. Solve the dierentzal equation
2x + y2 + 2:10yy' : 0.
ma: we NEW (mm m wwc.
Let
MA 266 Lecture 34
7.8 Repeated Eigenvalues
In this section, we consider the linear homogeneous system with constant coefcients
x = Ax
in which A has a repeated eigenvalue.
Example 1. Find the eigenvalues and eigenvectors of the following matrices
20 1~1
A
MA 266 Lecture 4
Section 2.2 Separable Equations
In this section, we use LI; to replace t as the independent variable.
The general form of a nonlinear rst order equation is
It can bewrittenin theform M( y 2) T N (X'a)%cfw_g? vv
If M is a function of w onl
MA 266 Lecture 29
6.6 The Convolution Integral
In this section, we introduce an important tool for Laplace transform, which is known as the
convolution.
Question: What is the inverse Laplace of H(s), if H(s) 2 F(3)G(3)?
I
it a M was
If F(5) = cfw_f(t)
MA 266 Lecture 27
6.4 Differential Equations with Discontinuous Forcing
Functions
In this section, we use Laplace transform to solve differential equations whose nonhomogeneous
term, or forcing function, is discontinuous.
Example 1. Solve the initial valu
MA 266 Lecture 31
7.3 Linear Dependence, Eigenvalues, Eigenvectors
Linear Dependence
A set of k: vectors x(1),- ,and x( ()k )is said to be ( \KAW d%md:0_nzz if
M watt A5926 OFMWWW Mew 04,0, tyne/t
WlUWG/S'FMWI' WWO WM
/ ) )
0,)5.p Cue) Ckx (b); 0
On the o
MA 266 Lecture 32
7.5 Homogeneous Linear Systems with Constant Coef-
cients
We consider the system of homogeneous linear equations With constant coefcients:
x' = AX.
Example 1. Find the general solution of
r- '6
f x, , 2x, a) [ )9 =09
v " ng _,
X 3 v>C>ze
MA 266 Lecture 26
6.3 Step Functions
The most interesting applications of the Laplace transform occur in the solution of linear differential
equations with discontinuous or impulsive forcing functions. In this section, we develop some
additiOnal propertie
MA 266 Lecture 3
Section 2.1 Linear Equations; Method of Integrating
Factors
In this section, we consider the rst order linear equation.
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o The general form is
o The standard form is at
z A
7% -r FHZDQW)
Sonietirnes we can solve an rst
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)
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s
13.
uc (t)f (t c)
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 # 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):
MA 266 Lecture 35
7.9 Nonhomogeneous Linear Systems
In this section, we consider the nonhomogeneous system
x = Ax + g(t)
The general solution can be expressed as
x = c1x(1)(t) + - - - + cnx()(t) + v(t)
where c1x(1)(t) + + cnx()(t) is the general solution
MA 266 Lecture 8
Section 2.4 Differences Between Linear and Nonlinear
Equations
Does every initial value problem have exactly one solution?
For a linear equation y + p(t)y 2 g(t) we have the following fundamental theorem
Theorem (linear equation) If the
MA 266 Lecture 24
6.1 Denition of the Laplace Transform
Many practical engineering problems involve mechanical or electrical systems acted on by discon
tinuous or impulsive forcing terms. The methods introduced in Chapter 3 are often rather awkward
to use
MA 266 Lecture 18
Section 3.6 Variation of Parameters
In this section, we consider a more generally applicable approach to nd particular solution
of nonhomogeneous equation. The method is called variation of parameters.
We will use the following example t
MA 266 Lecture 11
Section 2.7 Numerical Approximation: Eulers Method
In this section, we introduce a numerical method for solving the rst order initial value
problem
y()to -yo
The method is called ggd es iii:ZMM TM?Q1g:(t)? (2)32 [A
How to use tangent lin
MA 266 Lecture 13
Section 3.1 Homogeneous Equations with Constant
Coefcients
Terminologies
Starting from this section, we study the second order dierential equation of the form
Cl at
0V: 7" l I - Wt )
The above equation is called linear if
cfw_41,34 0% 23
MA 266 Lecture 16
Section 3.4 Repeated Roots; Reduction of Order
Review. Consider the linear homogeneous equation with constant coefficient
ay + by + cy : O.
The characteristic equation is
Wit in t (L t J
If the discriminant b2 4ac > 0, then
JVWQ Okt'thUi
MA 266 Lecture 25
6.2 Solution of Initial Value Problems
In this section, we show how the Laplace transform can be used to solve initial value problem for
linear differential equations with constant coefcients.
Review ( cfw_
The Laplace transform cfw_ f (
MA 266 Lecture 20
Section 3.7 Mechanic and Electrical Vibrations (contd)
Review For undamped free Vibrations, the governing equation is
WWW bike
Example 1. (Problem 6) A mass of 100 g stretches a, spring 50m. If the mass is set in
motion from its equilibr
MA 266 Lecture 5
Section 2.2 Separable Equations (contd)
Example 1. Consider the initial value problem
9 = tv(4 y)/3, W) = yo > 0.
(a).Dete7"mine how the behavior of the solution as t increases depends on the initial value yo.
(b).Suppose that yo = 0.5. F
MA 266 Lecture 14
Section 3.2 Solutions of Linear Homogeneous Equa-
tions; Wronskian
Terminologies 7'29 P00 y! 4, oi? 17;?
In this section, we study the structure of solutions of second order linear differential equation.
Let p and g be continuous functio
MA 266 Lecture 21
Section 3.8 Forced Vibration
In this section, we consider the situation in which a periodic external force is applied to a
spring-mass system.
Forced Vibration with Damping
Example 1. Suppose that the motion of a certain springmass syste
MA 266 Lecture 33
7.6 Complex Eigenvalues
In this section, we consider the system of linear homogeneous equations With constant coefcients.
We focus on the case that the coefcient matrix has complex eigenvalues.
Example 1. Find the general solution
/; K L
MA 266 Lecture 17
Section 3.5 Nonhomogeneous Equations; Method of
Undetermined Coefcients
We consider the nonhomogeneous equation
Lly] = y + le + 61(034 = W) 32"
The corresponding homogeneous equation is
ix:y~:3wP\-kakf irWQ go: 864?
The structure of the
MA 266 Lecture 2
Section 1.2 Solutions of Some Differential Equations
In this section, we discuss how to solve the differential equation of the following form
dy
E(JXLJb,
Where a and b are given constants.
Example 1. Find the solution v(t) to the followin
MA 266 Lecture 15
Section 3.3 Complex Roots of Characteristic Equation
Review. Consider the linear homogeneous equation with constant coefcient
ay" + by' + cy = 0.
The Characteristic equation is
Nat bf + c v; 0
If the discriminant b2 4ac > 0, then
ve 0+0,
MA 266 Lecture 6
Section 2.3 Modeling with First Order Equations
In this section, we consider several mathematical models using rst order differential equa
tions.
Steps in the process of mathematical modeling:
1. Construction of the Model 4ng
WWuw Moegvls
MA 266 Lecture 30
7.1 Introduction of System of First Order Linear System
In this chapter, we study the system of rst order linear differential equations.
Example 1. Rewrite the second order equations as a, system of rst order equations.
u + 0.125u + u =