Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 7.4: Basic Theory of Systems of First Order Linear
Equations
The general theory of a system of n first order linear equations
x1! = p11 (t ) x1 + p12 (t ) x2 + + p1n (t ) xn + g1 (t )
x2! = p21 (t ) x1 + p22 (t ) x2 + + p2 n (t ) xn + g 2 (t )
!
xn! =
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 7.6: Complex Eigenvalues
!
We consider again a homogeneous system of n first order
linear equations with constant, real coefficients,
x1! = a11 x1 + a12 x2 + + a1n xn
x2! = a21 x1 + a22 x2 + + a2 n xn
!
xn! = an1 x1 + an 2 x2 + + ann xn ,
and thus the
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 7.8: Repeated Eigenvalues
!
We consider again a homogeneous system of n first order
linear equations with constant real coefficients x' = Ax.
If the eigenvalues r1, rn of A are real and different, then
there are n linearly independent eigenvectors (1
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 10.1:TwoPoint Boundary Value Problems
In many important physical problems there are two or more
independent variables, so the corresponding mathematical
models involve partial differential equations.
This chapter treats one important method for solv
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 9 : Nonlinear Differential Equations and
Stability
Ch 9.1: The Phase Plane: Linear Systems
There are many differential equations, especially nonlinear
ones, that are not susceptible to analytical solution in any
reasonably convenient manner.
Numerica
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 9.3: Almost Linear Systems
In Section 9.1 we gave an informal description of the stability
properties of the equilibrium solution x = 0 of the 2 x 2 system
x' = Ax. The results are summarized in Table 9.1.1.
We required detA 0, and hence x = 0 is the
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 10.4: Even and Odd Functions
Before looking at further examples of Fourier series it is useful
to distinguish two classes of functions for which the EulerFourier formulas for the coefficients can be simplified.
The two classes are even and odd functi
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 7.1: Introduction to Systems of First Order
Linear Equations
A system of simultaneous first order ordinary differential
equations has the general form
x1! = F1 (t , x1 , x2 , xn )
x2! = F2 (t , x1 , x2 , xn )
!
xn! = Fn (t , x1 , x2 , xn )
where each
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 6.2: Solution of Initial Value Problems
The Laplace transform is named for the French mathematician
Laplace, who studied this transform in 1782.
The techniques described in this chapter were developed
primarily by Oliver Heaviside (18501925), an Eng
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 5.4: Euler Equations; Regular Singular Points
Recall that the point x0 is an ordinary point of the equation
d2y
dy
P( x) 2 + Q( x) + R( x) y = 0
dx
dx
if p(x) = Q(x)/P(x) and q(x)= R(x)/P(x) are analytic at at x0.
Otherwise x0 is a singular point.
Th
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 2.4: Differences Between Linear and Nonlinear Equations
Recall that a first order ODE has the form y' = f (t, y), and is
linear if f is linear in y, and nonlinear if f is nonlinear in y.
Examples: y' = t y  e t, y' = t y2.
In this section, we will
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 5.2: Series Solutions Near an Ordinary Point, Part I
!
!
In Chapter 3, we examined methods of solving second order
linear differential equations with constant coefficients.
We now consider the case where the coefficients are functions
of the independ
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 3.2: Fundamental Solutions of Linear Homogeneous
Equations
Let p, q be continuous functions on an interval I = (, ),
which could be infinite. For any function y that is twice
differentiable on I, define the differential operator L by
L[y] = y + p y +
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 1.1: Basic Mathematical Models;
Direction Fields
Differential equations are equations containing derivatives.
Ex>
Motion of fluids
Motion of mechanical systems
Flow of current in electrical circuits
Dissipation of heat in solid objects
Seismic waves
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 3.4: Repeated Roots; Reduction of Order!
!
Recall our 2nd order linear homogeneous ODE
ay! + by! + cy = 0
where a, b and c are constants.
Assuming an exponential soln leads to characteristic equation:
y(t ) = ert ar 2 + br + c = 0
Quadratic formula
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 2.9: First Order Difference Equations
!
!
Although a continuous model leading to a differential equation
is reasonable and attractive for many problems, there are some
cases in which a discrete model may be more appropriate.
Examples of this include a
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 4.3: Nonhomogeneous Equations: Method of
Undetermined Coefficients
The method of undetermined coefficients can be used to
find a particular solution Y of an nth order linear, constant
coefficient, nonhomogeneous ODE
L[y ] = a0 y ( n ) + a1 y ( n 1) +
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 4.1: Higher Order Linear ODEs: General
Theory!
!
An nth order ODE has the general form
dny
d n1 y
dy
P0 (t ) n + P1 (t ) n1 + ! + Pn1 (t ) + Pn (t ) y = G(t )
dt
dt
dt
We assume that P0, Pn, and G are continuous realvalued
functions on some interval
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 2.6: Exact Equations and Integrating Factors
Consider a first order ODE of the form
M ( x, y ) + N ( x, y ) y = 0
Suppose there is a function such that
x ( x, y ) = M ( x, y ), y ( x, y ) = N ( x, y )
and such that (x,y) = c defines y = (x) implicit
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
The Laplace Transform
Let f be a function defined for t 0, and satisfies certain
conditions to be named later.
The Laplace Transform of f is defined as an integral
transform:
Lcfw_ f (t ) = F ( s) = e st f (t )dt
0
The kernel function is K(s,t) = est.
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 2.1: Linear Equations; Method of Integrating Factors
A linear first order ODE has the general form
dy
= f (t , y )
dt
where f is linear in y. Examples include equations with
constant coefficients, such as those in Chapter 1,
y = ay + b
or equations wi
Ulsan National Institute of Science and Technology
Differential Equations
MATH 101

Spring 2017
Ch 6.1: Definition of Laplace Transform
Many practical engineering problems involve mechanical or
electrical systems acted upon by discontinuous or impulsive
forcing terms.
For such problems the methods described in Chapter 3 are
difficult to apply.
In
Ulsan National Institute of Science and Technology
power electronics
MATH EG 133

Spring 2017
Academic Integrity
Any act of plagiarism or cheating may
result in a 0 or negative grade on the
assignment OR an F in the class.
If you have any trouble on an assignment
or in this class, please come talk to me at
any time.
Power Electronics () ECE 404
Ulsan National Institute of Science and Technology
power electronics
MATH EG 133

Spring 2017
Online Lessons and Quizzes
1. Go to: http:/www.kedux.kr/
2. Login using student number () as
your ID and password ().
3. Click on to .
Power Electronics () ECE 404
11
Ulsan National Institute of Science and Technology
power electronics
MATH EG 133

Spring 2017
Attendance
Checked by automated attendance system
+2 point for ontime attendance
+1 for late attendance
0 for absence
(excused and unexcused are treated the same)
Power Electronics () ECE 404
14
Ulsan National Institute of Science and Technology
power electronics
MATH EG 133

Spring 2017
Fundamentals of Power Electronics:
Course Introduction
EE404: Fundamentals of
Power Electronics
()
Class Introduction
Power Electronics () ECE 404
2
Teaching Staff
Professor: Katherine A. Kim (,)
Email: [email protected]
Office: EB2 4019 (Open Door Pol
Ulsan National Institute of Science and Technology
power electronics
MATH EG 133

Spring 2017
Late Work Policy
5 LateDay Passes for Problem Sets
Each pass gives you a 24hour extension on
the deadline for one homework assignment.
If you run out of lateday passes, late
assignment are not accepted.
Each unused lateday pass is worth 1 point at