notes152-ch1 - MATH 152 Spring 2004-05 Applied Linear...

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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 January 29, 2005
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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 iii
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First-order differential equations (Chapter 2) y 0 = f ( t, y ) linear nonlinear integrating factor separable homogeneous exact integrating factor transform to separable transform to exact Second-order linear equations (Chapter 3) y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) homogeneous ⇐⇒ g 0 non-homogeneous ⇐⇒ g 6≡ 0 constant-coefficient non-constant-coefficient y = y h + y p characteristic equation 2 + + c = 0 trial and error y h y p λ 1 6 = λ 2 , both real λ 1 = λ 2 obtain y 1 λ 1 6 = λ 2 , both complex reduction of order to obtain y 2 variation of parameters no + constant-coeff? g ( t ) = P n ( t ) e αt cos βt + Q n ( t ) e αt sin βt undetermined coefficients
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Laplace transform (Chapter 4) L{ f ( t ) } LT Table Not in LT Table use definition of LT piecewise functions impulse functions complicated functions L - 1 { F ( s ) } Not in LT Table F ( s ) = F 1 ( s ) F 2 ( s ) F ( s ) = 1 as 2 + bs + c convolution completing square LT Table
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Chapter 1 Introduction 1.1 What are Differential Equations? A differential equation is an equation relating a function (unknown and to be solved ! ), its derivatives, and the variables upon which the function depends. For an example, d 2 f dt 2 + t df dt + f = 1 is a differential equation stating a relationship between the function f , its first and second derivatives (denoted by df dt , d 2 f dt 2 or simply we may use f 0 , f 00 for derivatives), and its independent variable t . Such an equation is assumed to hold not just at one point t but for a whole range of t , say for all points t lying between t = a and t = b . Thus, the equation above could be written more explicitly as d 2 f dt 2 ( t ) + t df dt ( t ) + f ( t ) = 1 , for all t [ a, b ] . (1.1) Since there are infinitely many distinct points t along the interval [ a, b ], one may say that Eq (1.1) provides an infinite number of equations. However, if we regard a differential equation as a set of an infinite number of equations, we must also content with the fact that we have an infinite number of unknowns; the values of f and its derivatives at each distinct point in [ a, b ]. Moreover, one must bear in mind that the derivative of a function f at a point t does not depend directly on its value at the point t but rather its values in a neighborhood of the point t . So the unknowns f ( t ), f 0 ( t ), f 00 ( t ), t [ a, b ], are not completely independent; nor is their interdependency easily expressed in terms of algebraic equation. In short, a differential equation cannot be solved by purely algebraic methods.
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