4.8 Nonhomogeneous Linear Systems
We now address the problem of finding the general solution to the nonhomogeneous
first order system
Notice a couple of changes:
1) There is a term g(t) that makes the system nonhomogeneous.
2) We have returned to the case
1.3 Direction Fields
Direction fields will help us to understand the differential equations of the
form:
The direction fields provide the insight into the qualitative behavior of solutions of a differential equation.
The Direction Field for a Differential
3.11 Higher Order Linear Homogeneous Differential
Equations
So far we have only studied second order equations. Equations of higher
order sometimes arise in mechanics: the fourth order equation models an
elastic beam under load. It turns out that higher o
2.4 Population Dynamics and Radioactive Decay
Population Models
No matter how big the population is, the following equation should hold:
Well call this the Conservation of Population Law.
Now, lets assume that the population is big enough to be modeled wi
2.6 Separable First Order Equations
What Is a Separable Equation?
Definition
A differentiable equation of the form:
is called a
Are the following equations separable?
1.
dy
dt
= 3y
2. y 0 =
5t+6
2y 2
3. y 0 = 3y + t
1
Solving a Separable Equation
Example
2.5 First Order Nonlinear Differential Equations
In sections 2.5 and 2.6 several types of nonlinear differential equations are
considered. In general, the first order nonlinear equation has the form:
Examples:
Existence and Uniqueness
Theorem 2.2
Let R be
3.1 Introduction to Second Order Differential Equations
Chapter 3 discusses second order linear differential equations:
What Is Different?
How many initial conditions are required to solve a second order equation?
Existence and Uniqueness
Theorem 3.1
Let
1.1 & 1.2 Introduction to and Examples of Differential
Equations
Introduction
In the real world most physical processes evolve according to the laws that
connect not only the current position of the object (or concentration of the
chemicals, or electrical
4.7 Repeated Eigenvalues
Example 1:
0 1 1
y0 = 1 0 1 y,
1 1 0
1 = 2, 2,3 = 1
So far, we have only encountered solutions that arise from eigenpairs. We
have never seen the situation of an eigenvalue with no matching eigenvector.
Algebraic and Geometric Mul
4.4 Constant Coefficient Homogeneous Systems;
The Eigenvalue Problem
In this section we develop a method to solve first order systems in the case
when the matrix P (t) is a constant matrix A
The Eigenvalue Problem
We look for solutions in the form:
We sub
3.5 Complex Roots
In this case the roots of characteristic polynomials are
For convenience we denote
and our solutions become:
We need to clarify the mathematical meaning of these two expressions. How
does the exponential function extend to accommodate co
3.2 The General Solution of Homogeneous Equations
Some Review of Linear Algebra
Definition
The set S = cfw_v1 , v2 , . . . vn is a Linearly Independent Set if
?
Theorem
The Following Are Equivalent: If A is an n n matrix
1.
2.
3.
4.
Recall:
1
4
7
1
2
3.8 The Method of Undetermined Coefficients
We can summarize the process of solving the Initial Value Problem
y 00 + p(t)y 0 + q(t)y = g(t), y(t0 ) = y0 , y 0 (t0 ) = y00
1. Find the general solution:
Find the complementary solution
Find any particular
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2.2 First Order Linear Differential Equations
Solving the Linear Homogeneous Equation
We start with rewriting the equation y 0 + p(t)y = 0 in the form:
Example 1: Find a solution of the equation y 0 + 3t2 y = 0
1
The General Solution
So far we have found
4.3 Homogeneous Linear Systems
We will develop methods for solving linear systems by going through the same
steps as we did before. First, we investigate the properties of homogeneous
systems. Then, we develop a method for solving homogeneous constant coe
2.3 Introduction to Mathematical Models
Now that we know how to solve linear ODEs, we can solve some application
problems!
Modeling Mixing Problems
Assume a salt solution enters a tank (which already contains a volume of fluid
with salt dissolved througho
2.9 Applications to Mechanics
Newtons Second Law:
What if we drop a ball? Assume there is no air resistance and the only force
is the weight of the object.
What if we have air resistance?
1
Case 1: Drag Force Proportional to Velocity
2
Note that the equat
4.1 Introduction to First Order Linear Systems
The differential equations we discussed in chapters 13 have enough power to
model the behavior of quite complicated phenomena. However, they all have
a fundamental disadvantage: they cannot model complex sys
4.6 Complex Eigenvalues
As the name of the section suggests, we are going to find solutions to the
system y0 = Ay, where A is a matrix with complex eigenvalues.
Example 1: Find the eigenvalues and eigenvectors
for the matrix A that
1 1
corresponds to the
3.12 Higher Order Homogeneous Constant Coefficient
Differential Equations
We have studied the method of finding a fundamental set of solutions for second order DEs with constant coefficients. The method is easily generalized
to the case of higher order eq