Ch7.6:ComplexEigenvalues
We consider again a homogeneous system of n first order
linear equations with constant, real coefficients,
x1 a11 x1 a12 x2 a1n xn
x a21 x1 a22 x2 a2 n xn
2
x an1 x1 an 2 x2 ann xn ,
n
and thus the system can be written as x' = Ax
Ch3.1:SecondOrderLinearHomogeneous
EquationswithConstantCoefficients
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
Ch 3.3:
Complex Roots of Characteristic Equation
Recall our discussion of the equation
ay by cy 0
where a, b and c are constants.
Assuming an exponential soln leads to characteristic
equation:t ) e rt ar 2 br c 0
y(
Quadratic formulab(or factoring) yields
Ch2.2:SeparableEquations
In this section we examine a subclass of linear and nonlinear
first order equations. Consider the first order equation
dy
f ( x, y )
dx
We can rewrite this in the form
M ( x, y ) N ( x, y )
dy
0
dx
For example, let M(x,y) = - f (
Ch3.5:NonhomogeneousEquations;
MethodofUndeterminedCoefficients
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
Ch3.6:VariationofParameters
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 variation
Ch3.4:
RepeatedRoots;ReductionofOrder
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:t ) e rt ar 2 br c 0
y(
Quadratic formulab factoring) yields two sol
Ch2.6:
ExactEquations&IntegratingFactors
Consider a first order ODE of the form
M ( x, y ) N ( x, y ) y0
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) implicitly. Then
M ( x, y
Ch1.3:
ClassificationofDifferentialEquations
The main purpose of this course is to present methods of
finding solutions, and to discuss properties of solutions of
differential equations.
To provide a framework for this discussion, in this section
we give
Ch1.2:
SolutionsofSomeDifferentialEquations
Recall the free fall and owl/mice differential equations:
v 9.8 0.2v,
p0.5 p 450
These equations have the general form y' = ay - b
We can use methods of calculus to solve differential
equations of this form.
Exa
Ch1.1:
BasicMathematicalModels;DirectionFields
Differential equations are equations containing derivatives.
Derivatives describe rates of change.
The following are examples of physical phenomena
involving rates of change:
Motion of fluids
Motion of mechan
Homework 4
Problems 1 through 5: Find the solution of the given initial value problem.
1. y ' y = 2 xe 2 x with y (0) = 1 .
2. y '+ 2 y = xe 2 x with y (1) = 0 .
3. xy '+ 2 y = x 2 x + 1 with y (1) = 1/ 2 , x > 0 .
4. y '+ (2 / x) y = (cos x) / x 2 with y
Ch5.1:ReviewofPowerSeries
Finding the general solution of a linear differential equation
depends on determining a fundamental set of solutions of the
homogeneous equation.
So far, we have a systematic procedure for constructing
fundamental solutions if eq
Ch3.2:SolutionsofLinearHomogeneousEquations;Wronskian
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 q y
Note that L[
Ch7.5:HomogeneousLinearSystemswith
ConstantCoefficients
We consider here 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
This system can be
Ch7.4:BasicTheoryofSystemsofFirst
OrderLinearEquations
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 )
x pn1 (t ) x1 pn 2 (t ) x2 pnn (t ) xn
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 in
Ch6.4:DifferentialEquationswith
DiscontinuousForcingFunctions
In this section, we focus on examples of nonhomogeneous
initial value problems in which the forcing function is
discontinuous.
ay by cy g (t ), y 0 y0 , y 0 y0
Example 1: Initial Value Problem
Ch6.2:SolutionofInitialValueProblems
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 (1850-1925), an English
ele
Ch6.3:StepFunctions
Some of the most interesting elementary applications of the
Laplace Transform method occur in the solution of linear
equations with discontinuous or impulsive forcing functions.
In this section, we will assume that all functions consid
Ch6.1:DefinitionofLaplaceTransform
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 this ch
Ch5.5:SeriesSolutions
NearaRegularSingularPoint,PartI
We now consider solving the general second order
linear equation in the neighborhood of a regular
singular point x0. For convenience, will will take x0 =
0.
The point x0 = 0 is a regular singular point
Ch5.4:EulerEquations;
RegularSingularPoints
Recall that for equation
d2y
dy
P ( x) 2 Q ( x) R ( x) y 0
dx
dx
if P, Q and R are polynomials having no common factors,
then the singular points of the differential equation are the
points for which P(x) = 0.
E
Ch5.2:SeriesSolutions
NearanOrdinaryPoint,PartI
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 variable,
Ch2.1:LinearEquations;
MethodofIntegratingFactors
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 with variable
Homework 5
Find the solution y = y ( x) of the given initial value problems.
1.
y '+ y ' 2 y = 0,
y (0) = 1,
y '(0) = 1
2.
y '+ 4 y '+ 3 y = 0,
y (0) = 2,
y '(0) = 1
3.
y '+ 3 y ' = 0,
y (0) = 2,
y '(0) = 3
4.
y '+ 4 y = 0,
y (0) = 0,
y '(0) = 1
5.
y '+ 4
Homework 2
By separating variables, find the solution of the following initial value problems in
explicit form:
1.
y ' = 2 x / ( y + x 2 y ),
y (0) = 2.
2.
y ' = xy 3 (1 + x 2 ) 1/2 ,
y (0) = 1.
3.
y ' = 3 x 2 / (3 y 2 4),
(Leave the solution as a cubic e
Homework for Week 1
1. Find the derivatives of the both sides of the addition formula
sin( x + x0 ) = sin x cos x0 + cos x sin x0 with x0 being an arbitrary constant. This gives
you the other addition formula for cos( x + x0 ) .
Solution:
cos( x + x0 ) =
Solutions
Math 2351Fall 2012
Week 12 Worksheet: Systems of ODEs (Part 2)
1. (7.6, p. 390, problem 2, 3, 5) For the following matrices A, nd the general solution of x = Ax,
compute L = x1 x2 x2 x1 and its sign, and state whether the phase-space trajectorie
Solutions
Math 2351Fall 2012
Week 11 Worksheet: Euler Equations and Systems of ODEs (Part 1)
(Partial solution of this worksheet will be available at your tutors website.)
1. (5.5, p165, problem 1, 3, 11) Determine the general solution of the given dieren