Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Section 3.6: Variation of Parameters
I
Recall the nonhomogeneous equation
y 00 + p(t)y 0 + q(t)y = g(t).
I
where p,q, and g are continuous functions on an open
interval I.
The associated homogeneous equation is
y 00 + p(t)y 0 + q(t)y = 0.
I
I
In this sec
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
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 equation
MTH 201

Fall 2016
Ch 7.2: Review of Matrices
For theoretical and computational reasons, we review results
of matrix theory in this section and the next.
A matrix A is an m x n rectangular array of elements,
arranged in m rows and n columns, denoted
a11
a21
A ai j
a
m
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch6.5:ImpulseFunctions
In some applications, it is necessary to deal with phenomena
of an impulsive nature.
For example, an electrical circuit or mechanical system subject
to a sudden voltage or force g(t) of large magnitude that acts
over a short time
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 5.5: Series Solutions Near a Regular Singular
Point, I
We now consider solving the general second order linear
equation in the neighborhood of a regular singular point
x0. For convenience, we will take x0 = 0.
Recall that the point x0 = 0 is a regula
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 6.4: Differential Equations with Discontinuous
Forcing Functions
In this section focus on examples of nonhomogeneous initial
value problems in which the forcing function is discontinuous.
ay by cy g (t ), y0 y0 , y0 y0
Example 1: Initial Value Problem
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 7.3: Systems of Linear Equations, Linear
Independence, Eigenvalues
A system of n linear equations in n variables,
a1,1 x1 a1, 2 x2 a1,n xn b1
a2,1 x1 a2, 2 x2 a2,n xn b2
an ,1 x1 an , 2 x2 an ,n xn bn ,
can be expressed as a matrix equation Ax = b:
a
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
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.
Thu
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 7.7: Fundamental Matrices
Suppose that x(1)(t), x(n)(t) form a fundamental set of
solutions for x' = P(t)x on < t < .
The matrix
x1(1) (t ) x1( n ) (t )
(t )
,
x (1) (t ) x ( n ) (t )
n
n
whose columns are x(1)(t), x(n)(t), is a fundamental ma
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 7.9: Nonhomogeneous Linear Systems
The general theory of a nonhomogeneous system of 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 pn1 (t ) x1 pn 2 (t ) x2 pnn (t ) xn g n (t )
parallels that
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 7.5: Homogeneous Linear Systems with Constant
Coefficients
We consider here a homogeneous system of n first order linear
equations with constant, real coefficients:
x1 a11x1 a12 x2 a1n xn
x2 a21x1 a22 x2 a2 n xn
xn an1 x1 an 2 x2 ann xn
This system c
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 7.6: Complex Eigenvalues
We consider again a homogeneous system of n first order
linear equations with constant, real coefficients,
x1 a11x1 a12 x2 a1n xn
x2 a21x1 a22 x2 a2 n xn
xn an1 x1 an 2 x2 ann xn ,
and thus the system can be written as x' = Ax
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 3.1: 2nd Order Linear Homogeneous
EquationsConstant Coefficients
A second order ordinary differential equation has the
general form
y f (t, y, y)
where f is some given function.
This equation is said to be linear if f is linear in y and y':
y g (t )
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 4.1: Higher Order Linear ODEs:
General Theory
An nth order linear ODE has the general form
dny
d n1 y
dy
P0 (t ) n P1 (t ) n 1 Pn1 (t ) Pn (t ) y Gt
dt
dt
dt
We assume that P0, Pn, and G are continuous realvalued
functions on some interval I = (, )
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Section 2.5: Autonomous Equations and Population
Dynamics
I
I
In this section we examine equations of the form y 0 = f (y ),
called autonomous equations, where the independent
variable t does not appear explicitly.
I
The main purpose of this section is to
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Section 2.6 Exact Equations and Integrating Factors
Exact equation
Consider a first order ODE of the form
M(x, y ) + N(x, y )y 0 = 0.
This equation is called exact, if there is a function
x (x, y )
= M(x, y ),
y (x, y )
= N(x, y ).
such that
Exact equatio
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Section 3.1: 2nd Order Linear Homogeneous
EquationsConstant Coefficients
I
A second order ordinary differential equation has the general
form
y 00 = f (t, y , y 0 )
where f is some given function.
I
This equation is said to be linear if f is linear in y
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 2.5: Autonomous Equations and
Population Dynamics
In this section we examine equations of the form dy/dt = f
(y), called autonomous equations, where the independent
variable t does not appear explicitly in f. (so separable!)
The main purpose of this
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 3.6: Variation of Parameters
Recall the nonhomogeneous equation
y p(t ) y q(t ) y g (t )
where p, q, g are continuous functions on an open interval I.
The associated homogeneous equation is
y p(t ) y q(t ) y 0
In this section we will learn the varia
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Differential Equations  Practice Problems
Section 5.5. Series Solutions Near a Regular Singular Point
Problem 51. Consider the following equation
2x2
d2 y
dy
+ x (1 + x)y = 0.
2
dx
dx
(a) Show that x = 0 is not a regular point, but a regular singular po
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Review for Sections 2.1 & 2.2
Solving 1st order linear ODEs by integrating
factor method: dy p(t ) y g (t ), (t ) e p (t ) dt .
dt
: After multiplying and simplifying, integrate
and solve for y
Solving 1st order separable ODEs:
M ( x)dx N ( y)dy 0
: Dir
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 3.5: Nonhomogeneous Equations: Method of
Undetermined Coefficients
Return to the nonhomogeneous (2nd order linear) ODE
y p(t ) y q(t ) y g (t )
where p, q, g are continuous functions on an open interval I.
The associated homogeneous equation is
y p(t
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Review of Ch.1
DEs are equations with derivatives.
Classification of DEs by the following :
ODE vs PDE
Order of DEs
Linear vs nonlinear
Issues for solutions of DEs: existence,
uniqueness and how to find it.
Direction field, equilibrium solutions, syst
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
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 independent