notes152-ch2

notes152-ch2 - MATH 152 Spring 2004-05 Applied Linear...

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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 January 29, 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 iii Chapter 2 First-Order Differential Equations We now turn our attention to the problem of constructing explicit analytic solutions of differential equa- tions; that is to say, solutions that can be expressed in terms of elementary functions (or formulae). Such construction of solutions always involves different mathematical techniques. Let us first consider the case of first-order linear differential equations. 2.1 First-Order Linear Differential Equations A first-order linear (ordinary) differential equation has the form a ( t ) y + b ( t ) y = c ( t ) . (2.1) Here y represents the unknown function, y its derivative with respect to the (one and only one) independent variable t , and a ( t ), b ( t ) and c ( t ) are certain prescribed functions of t . As long as a ( t ) 6 = 0, Eq (2.1) is equivalent to a differential equation of the form y + p ( t ) y = g ( t ), (2.2) by regarding p ( t ) = b ( t ) /a ( t ) and g ( t ) = c ( t ) /a ( t ). We shall refer to a differential equation (2.2) as the standard form of differential equation (2.1). In general, we shall say that an ordinary differential equation is in standard form when the coefficient of the highest derivative is 1. Our goal now is to develop a formula for the general solution of Eq (2.2). To accomplish this goal, we shall first construct solutions in several examples. Then with the knowledge gained from these simpler examples, we shall present a nice formula for the solution of any differential equations of the form (2.2). ¥ Example 2.1.1 (First-order linear equation) Solve the differential equation y- λy = 0 , λ a real constant . (2.3) Solution We multiply e- λt to Eq (2.3) and get ‡ e- λt y · = e- λt y- λe- λt y = e- λt ( y- λy ) = 0 ....
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notes152-ch2 - MATH 152 Spring 2004-05 Applied Linear...

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