notes152-ch4

# notes152-ch4 - MATH 152 Spring 2004-05 Applied Linear...

This preview shows pages 1–6. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 152 Spring 2004-05 Applied Linear Algebra & Differential Equations Lecture Notes Dr. Tony Yee Department of Mathematics The Hong Kong University of Science and Technology March 1, 2005 ii Contents Table of Contents iii 1 Introduction 3 1.1 What are Differential Equations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Classification of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Solutions of Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 5 2 First-Order Differential Equations 11 2.1 First-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 The Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Modeling with First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Euler and Runge–Kutta Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . 35 3 Second-Order Linear Equations 43 3.1 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.1 Homogeneous Equations with Real Constant Coefficients . . . . . . . . . . . 46 3.1.2 Method of Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Nonhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.1 Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . 55 3.2.2 Method of Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3 Mechanical Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4 Laplace Transform 81 4.1 Introduction – Examples and Properties . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 Initial Value Problems and Inverse Laplace Transform . . . . . . . . . . . . . . . . . 86 4.3 Discontinuous Forcing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Impulse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 iii Chapter 4 Laplace Transform 4.1 Introduction – Examples and Properties In this chapter we will be looking at how to use Laplace transforms to solve differential equations. There are many kinds of transforms out there in the world. Laplace transforms and Fourier transforms are probably the main two kinds of transforms that are used. As we will see in the following we can use Laplace transforms to reduce a differential equation to an algebra problem. The algebra can be messy on occasion, but it will be simpler than actually solving the differential equation directly in many cases. Laplace transforms can also be used to solve a wide variety of initial value problems that we cannot use any previous methods on.also be used to solve a wide variety of initial value problems that we cannot use any previous methods on....
View Full Document

{[ snackBarMessage ]}

### Page1 / 26

notes152-ch4 - MATH 152 Spring 2004-05 Applied Linear...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online