MA441 - CH02 - im02.qxd 10:57 AM Page 30 CHAPTER 2...

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CHAPTER 2 Second-Order Linear ODEs Major Changes Among linear ODEs those of second order are by far the most important ones from the viewpoint of applications, and from a theoretical standpoint they illustrate the theory of linear ODEs of any order (except for the role of the Wronskian). For these reasons we consider linear ODEs of third and higher order in a separate chapter, Chap. 3. The new Sec. 2.2 combines all three cases of the roots of the characteristic equation of a homogeneous linear ODE with constant coefficients. (In the last edition the complex case was discussed in a separate section.) Modeling applications of the method of undetermined coefficients (Sec. 2.7) follow immediately after the derivation of the method (mass–spring systems in Sec. 2.8, electric circuits in Sec. 2.9), before the discussion of variation of parameters (Sec. 2.10). The new Sec. 2.9 combines the old Sec. 1.7 on modeling electric circuits by first-order ODEs and the old Sec. 2.12 on electric circuits modeled by second-order ODEs. This avoids discussing the physical aspects and foundations twice. SECTION 2.1. Homogeneous Linear ODEs of Second-Order, page 45 Purpose. To extend the basic concepts from first-order to second-order ODEs and to present the basic properties of linear ODEs. Comment on the Standard Form (1) The form (1), with 1 as the coefficient of y , is practical, because if one starts from ƒ( x ) y g ( x ) y h ( x ) y r ( x ), one usually considers the equation in an interval I in which ƒ( x ) is nowhere zero, so that in I one can divide by ƒ( x ) and obtain an equation of the form (1). Points at which ƒ( x ) 0 require a special study, which we present in Chap. 5. Main Content, Important Concepts Linear and nonlinear ODEs Homogeneous linear ODEs (to be discussed in Secs. 2.1 2.6) Superposition principle for homogeneous ODEs General solution, basis, linear independence Initial value problem (2), (4), particular solution Reduction to first order (text and Probs. 15 22) Comment on the Three ODEs after (2) These are for illustration, not for solution, but should a student ask, answers are that the first will be solved by methods in Sec. 2.7 and 2.10, the second is a Bessel equation (Sec. 5.5) and the third has the solutions c 1 x c 2 with any c 1 and c 2 . Comment on Footnote 1 In 1760, Lagrange gave the first methodical treatment of the calculus of variations. The book mentioned in the footnote includes all major contributions of others in the field and made him the founder of analytical mechanics. 30 im02.qxd 9/21/05 10:57 AM Page 30
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Comment on Terminology p and q are called the coefficients of (1) and (2). The function r on the right is not called a coefficient, to avoid the misunderstanding that r must be constant when we talk about an ODE with constant coefficients . SOLUTIONS TO PROBLEM SET 2.1, page 52 2. cos 5 x and sin 5 x are linearly independent on any interval because their quotient, cot 5 x , is not constant. General solution: y a cos 5 x b sin 5 x .
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