ECE210 Lecture 2: notes
9 May 2018
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Contents
1 Introduction 1
1.1 Modelling with Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 First Order Equations 6
2.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Integrating Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Second Order Equations 18
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Constant Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Variable Coefficient Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Series Solutions 40
4.1 Review of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Solution Near Ordinary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Laplace Transforms 49
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 Inverse Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Applications to Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.5 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.6 The Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6 Partial Differential Equations 69
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Derivation of the Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 Linear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.4 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.5 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.5.1 Even, Odd and Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.5.2 Orthogonal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.5.3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.5.4 Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
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Math 201 CONTENTS ii
6.6 Separation of Variables (revisited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.6.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.6.2 Nonhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.6.3 Further Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.6.4 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Chapter 1
Introduction
1.1 Modelling with Differential Equations
Many fundamental principles or laws governing the behaviour of the natural world are expressed in terms
of certain quantities and their rates of change. In mathematical terms these expressions take the form of
equations involving these quantities and their derivatives. An equation involving some unknown function and
its derivatives is called a differential equation. A differential equation, or perhaps a system of differential
equations, that describe some physical process is often called a mathematical model of the process. In
order to understand these physical processes requires an understanding of, and an ability to solve, differential
equations.
Below we discuss a few physical problems and the mathematical models that govern them.
Newton’s Second Law of Motion
Everyone is familiar with Newton’s second law of motion which states that force acting on an object is equal
to the product of its mass and acceleration. This is represented mathematically by the following simple
equation:
F=ma. (1.1)
Here Frepresents the net force acting on the object, mrepresents mass of the object and aits acceleration.
If we denote the velocity of the object by v, then it is related to acceleration by a=dv/dt.
Apply Newton’s second law of motion to an object of mass mas it falls through the atmosphere. There are
two forces acting on the object: gravity and air resistence, sometimes called drag. The force of gravity is
equal to the weight of the object. The drag force is more difficult to model but it is often assumed to be
proportional to velocity. Let us denote by Fgand Fdthe forces of gravity and drag acting on the object.
Then
Fg=mg, and Fd=−kv,
where gis acceleration due to gravity (approximately 9.8m/sec2), and kis a positive constant called the
drag coefficient. The minus sign in the drag force reflects the fact that the drag force acting on an object is
always in the direction opposite to its velocity. The net force acting on the object is F=Fg+Fd=mg −kv.
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Document Summary
Many fundamental principles or laws governing the behaviour of the natural world are expressed in terms of certain quantities and their rates of change. In mathematical terms these expressions take the form of equations involving these quantities and their derivatives. An equation involving some unknown function and its derivatives is called a di erential equation. A di erential equation, or perhaps a system of di erential equations, that describe some physical process is often called a mathematical model of the process. In order to understand these physical processes requires an understanding of, and an ability to solve, di erential equations. Below we discuss a few physical problems and the mathematical models that govern them. Everyone is familiar with newton"s second law of motion which states that force acting on an object is equal to the product of its mass and acceleration. This is represented mathematically by the following simple equation:
