Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 2 - Linear, Constant-Coeff

Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 2 - Linear, Constant-Coeff

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CHAPTER 2 Linear, Constant-Coefcient Systems T here are few classes of ode s for which exact, analytical solutions can be obtained by hand. However, for many systems which cannot be solved explicitly, we may approximate the dynamics by using simpler systems of ode s which can be solved exactly. This often allows us to extract valuable qualitative information about complicated dynamical systems. We now introduce tech- niques for systematically solving linear systems of Frst-order ode s with constant coefFcients. Notation. Because we will be working with vectors of dependent variables, we should establish (or recall) some commonly used notation. We denote the set of real numbers by R . We let R n denote the set of all vectors with n components, each of which is a real number. Usually, vectors will be denoted by bold letters such as x , y , and we will use capital letters such as A to denote n × n matrices of real numbers. Generally, we shall not distinguish between row vectors and column vectors, as our intentions will usually be clear from the context. ±or example, if we write the product x A , then x should be treated as a row vector, whereas if we write A x , then x is understood to be a column vector. If we write x ( t ) , we mean a vector of functions, each of which depends on a variable t . In such cases, the vector x ( 0 ) would be a constant vector in which each component function has been evaluated at t = 0. Moreover, the vector x ± ( t ) is the vector consisting of the derivatives of the functions which form the components of x ( t ) . Systems with constant coefFcients. Suppose that y 1 , y 2 , . . . y n are variables which depend on a single variable t . The general form of a linear, constant- 8
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linear , constant - coefficient systems 9 coefFcient system of Frst-order ode s is as follows: d y 1 d t = a 11 y 1 ( t )+ a 12 y 2 ( t · · · + a 1 n y n ( t f 1 ( t ) d y 2 d t = a 21 y 1 ( t a 22 y 2 ( t · · · + a 2 n y n ( t f 2 ( t ) . . . d y n d t = a n 1 y 1 ( t a n 2 y 2 ( t · · · + a nn y n ( t f n ( t ) . ( 2 . 1 ) Here, each a ij is a constant ( 1 i , j n ) , and f i ( t ) ( i = 1, 2, . . . n ) are functions of t only. Example 2 . 0 . 3 . Soon, we will learn how to solve the linear, constant-coefFcient system d y 1 d t = 3 y 1 - 2 y 2 + cos t d y 2 d t = 10 y 2 - t 2 + 6. ( 2 . 2 ) The system ( 2 . 1 ) can be written more compactly if we introduce matrix/vector notation. Suppressing the dependence on t for notational convenience, routine matrix multiplication will verify that y ± 1 y ± 2 . . . y ± n = a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a n 1 a n 2 . . . a y 1 y 2 . . . y n + f 1 f 2 . . . f n ( 2 . 3 ) is equivalent to the system ( 2 . 1 ). ±urthermore, if we deFne y = y 1 y 2 .
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This note was uploaded on 02/27/2012 for the course MATH 532 taught by Professor Reynolds during the Fall '11 term at VCU.

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Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 2 - Linear, Constant-Coeff

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