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Title: linear ALgebra and ODES
Description: Engineering and Mechanics. Learning Mechanics. Fundamental Concepts. Units. Newtonian Gravitation.
Description: Engineering and Mechanics. Learning Mechanics. Fundamental Concepts. Units. Newtonian Gravitation.
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Linear Algebra and Ordinary Differential Equations
MTH-121
February 6, 2012
MTH-121
Linear Algebra and Ordinary Differential Equations
REVIEW AND INTRODUCTION
Before embarking on a study of differential equations, it is essential
to solidify your technical facility with differentiation and
integration
...
MTH-121
Linear Algebra and Ordinary Differential Equations
REVIEW AND INTRODUCTION
Differentiation:
Given a function y = f (x), its derivative, denoted by y or dy /dx ,
gives the instantaneous rate of change of f with respect to x
...
MTH-121
Linear Algebra and Ordinary Differential Equations
REVIEW AND INTRODUCTION
The second derivative identifies the concavity of the curve
y = f (x)
...
A curve y = f (x) is concave up at those points x where the second
derivative is positive, and concave down where the second
derivative is negative
...
MTH-121
Linear Algebra and Ordinary Differential Equations
REVIEW AND INTRODUCTION
MTH-121
Linear Algebra and Ordinary Differential Equations
REVIEW AND INTRODUCTION
Table 1 below lists the most frequently used properties of
derivatives and, for later reference, the corresponding properties of
integrals
...
Although Table 2 below does not contain every
differentiation formula, it will probably suffice to differentiate
almost every function you are likely to encounter in practice
...
(Note : To avoid the
repetition of writing “+ c” after every result in the right-hand
column, the arbitrary additive constant c has been omitted from
each of the integration formulas, as in Table 1 above
...
The
emergence of low-cost, high-speed computers has spawned new
techniques for solving differential equations, which allows problem
solvers to model and solve complex problems based on systems of
differential equations
...
If there is just one independent variable, then the derivatives are
all ordinary derivatives, and the equation is an ordinary
differential equation
...
(1)
dt 2
Another example is Bessel’s equation
x2
d 2y
dy
+x
+ (x 2 − n2 )y = 0,
2
dx
dx
(2)
where x is the independent variable, y is the dependent variable,
and n2 is a constant
...
Some common examples are the heat
equation
∂2u
∂u
=
,
∂t
∂x 2
(3)
1 ∂2u
∂2u
=
,
2 ∂t 2
c
∂x 2
(4)
∂2u ∂2u
+
= 0
...
MTH-121
Linear Algebra and Ordinary Differential Equations
Differential Equations: Basic Definitions
MTH-121
Linear Algebra and Ordinary Differential Equations
Differential Equations: Basic Definitions
MTH-121
Linear Algebra and Ordinary Differential Equations
Differential Equations: Basic Definitions
Consider the differential equation
MTH-121
Linear Algebra and Ordinary Differential Equations
Differential Equations: Basic Definitions
MTH-121
Linear Algebra and Ordinary Differential Equations
Differential Equations: Basic Definitions
MTH-121
Linear Algebra and Ordinary Differential Equations
Differential Equations: Basic Definitions
MTH-121
Linear Algebra and Ordinary Differential Equations
Differential Equations: Basic Definitions
MTH-121
Linear Algebra and Ordinary Differential Equations
Differential Equations: Basic Definitions
The order of a differential equation is the order of the highest
derivative which appears in the equation
...
The heat, wave, and
Laplace equations are examples of second order partial differential
equations
...
For virtually every such equation encountered
in practice, the general solution will contain one arbitrary constant,
that is, one parameter, so a first–order IVP will contain one initial
condition
...
MTH-121
Linear Algebra and Ordinary Differential Equations
Separable Equations
MTH-121
Linear Algebra and Ordinary Differential Equations
Separable Equations
MTH-121
Linear Algebra and Ordinary Differential Equations
Separable Equations
MTH-121
Linear Algebra and Ordinary Differential Equations
Separable Equations
MTH-121
Linear Algebra and Ordinary Differential Equations
Homogeneous Equations
MTH-121
Linear Algebra and Ordinary Differential Equations
Homogeneous Equations
MTH-121
Linear Algebra and Ordinary Differential Equations
Homogeneous Equations
MTH-121
Linear Algebra and Ordinary Differential Equations
Homogeneous Equations
MTH-121
Linear Algebra and Ordinary Differential Equations
Homogeneous Equations
MTH-121
Linear Algebra and Ordinary Differential Equations
Homogeneous Equations
MTH-121
Linear Algebra and Ordinary Differential Equations
Homogeneous Equations
MTH-121
Linear Algebra and Ordinary Differential Equations
Title: linear ALgebra and ODES
Description: Engineering and Mechanics. Learning Mechanics. Fundamental Concepts. Units. Newtonian Gravitation.
Description: Engineering and Mechanics. Learning Mechanics. Fundamental Concepts. Units. Newtonian Gravitation.