Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267 DifferentialEquations
Exam 2 Review
The following is a list of example problems from each section that will be covered in the second exam. In
order to fully use this review sheet, you should attempt all problems on your own. You should consult th
FINAL EXAM
Fall 267
Circle the correct answer. If you circle more than one answer the
credit for a problem is zero.
Problem 1.(10 POINTS.)Solution y (x) to the initial value problem
(x + 2xy )
dy
= (x2 + y 2 + y ),
dx
y (1) = 1
satises the equation:
x3
7
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267 DifferentialEquations
Exam 2 Review
The following is a list of example problems from each section that will be covered in the second exam. In
order to fully use this review sheet, you should attempt all problems on your own. You should consult th
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
MATH 267 Test N0. 3 SPRING 2016, Sections 1419, 2628
Problem 1(20 Points)
Calculate the Laplace Transform of the function f = f (t) Which is equal to
e3t when t is 1 _<_ t < 2 and equal to O for any other value of t.
Problem 2(20 points)
Calculate the inv
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
0 l 53 , \5
010% F'sxtb
9f: X1333
3x ,
: X;'bt:)/
So oWhMt CS MOI: _
IMOMMIT If m (11%me is car
9m; M, 44, HM, cfw_mufn FzFZn 3) i
Hun we com ekfRCh H erl
g
VhM H1.
(rm. 0. 1' M 30 Lab 8/ F= 13"5403 <0
a
W1 (1.0 cfw_not Knot, r
if . 7" x
)3
From 6) if: 1
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267: Elementary Differential Equations & Laplace Transform
8.2 Homogeneous linear systems
In this section, we Will learn to solve the homogeneous system of first order linear
equations With constant coefcient of the form:
,.e~* ~\
@l
Where
931
an
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267: Elementary Differential Equations & Laplace Transform
Chapter 7 : The Laplace transform
In this chapter, we study one special type of integral transforms: the Laplace
transform. We then apply the Laplace transform to solve differential equatio
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
 (kl (5(4) /
QMOCWj + QAc(ll.lj 4. _f 470/7 7641043 3767
Math 267: Elementary Differential Equations 8; Laplace Transform
4.3. Homogeneous linear equations with constant W
(:0 Q ff! C(eJS.
In this section, we learn a method for solving the homogeneous
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267: Elementary Differential Equations 85 Laplace Transform
2.5 Solving differential equations
by substitutions
We have learned three types of rstorder DES: separable equations, linear equations,
and exact equations. Given a DE of one of these types,
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
3.1 Applications of rstorder
DES: Linear models
(See the lecture note of section 1.3 for physical laws that we use in this section.)
Example 1. (Carbon dating)
Determine the approximate age of a piece of burned wood, if it was found that
(Sgggiof the C14
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
3.2 Applications of rstorder
DES: Nonlinear models
Logistic model in population dynamics:
In a population model, it is usually the case that the environment is not capable of
sustaining too many individuals, says, no more tha K individuals since eventual
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267: Elementary Differential Equations & Laplace Transform
2.4 Exact Equations
In this section, we will solve rstorder equations written in the form:
Noe the special way we use to write the equation. Exact equations are equations
when the left hand s
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267: Elementary Differential Equations 85 Laplace Transform
Chapter 8: Systems of linear differential equations
So far, we have learned several methods for solving a single differential equation.
In this chapter, we will learn systems of equations. T
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267: Elementary Differential Equations 85 Laplace remsforrn
7.4 Operational Properties II
In section 7.3, we found the Laplace transforms of a function f (t) multiplied by
6 or U (15 a), where U (t a) is the step unit function (Heaviside function). I
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267: Elementary Differential Equations & Laplace Transform
8.3 Nonhomogeneous linear systems
In this section, we will learn to solve nonhomogeneous system of rst order linear
equations with constant coefcients. The general form is
X = AX + F(t)
Whe
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Practice Final (266)
Supplemental Instruction
Iowa State University
Leader:
Course:
Instructor:
Date:
Matt E
Math 266/7
Various
42812
Supplemental Instruction
1060 HixsonLied Student Success Center 2946624 www.si.iastate.edu
Write the system of equati
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267: Elementary Differential Equations & Laplace Transform
1.2 Initial Value Problems (IVPS)
Recall: an nth order DE in the normal form:
(Tb1) 1,
<1Ei/
A ' c ')(.
Example 1. First order equation: 3/ = 225 has a onefamily of solutions (21 j? )L I
M
W
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267: Elementary Differential Equations & Laplace Transform
1.1: Denitions 8!. Terminology
Denition. A 11E is an equation which contains the derivatives of one or more
",3
unknown functions (dependent variables) an one or more independent variables.
Q
Elementary Differential Equations and Laplace Transforms.
MATH 267

Spring 2014
Math 267: Elementary Differential Equations & Laplace Transform
Chapter 4: Highorder linear equations
4.1. Concepts
Recall that an nthorder linear equation is written in the form:
an (x)y (n) + an1 (x)y (n1) + + a1 (x)y 0 + a0 (x)y = g(x).
4.1.1 Initial