1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 343 The following facts are proven in linear algebra. If the determinant of ...

344 Ordinary Differential Equations taneous homogeneous system of equations C1Y11 + C2Y12 + · · · + CnYln = 0 C1Y21 + C2Y22 + · ...

Linear Systems of First-Order Differential Equations 345 SOLUTION Forming the matrix Y whose jth column is Yj and computing det ...

346 Ordinary Differential Equations linearly independent on an interval, it may or may not be linearly independent at each point ...

Linear Systems of First-Order Differential Equations 347 EXAMPLE 2 Verifying Linear Independence of a Set of Vector Functions Sh ...

348 Ordinary Differential Equations In exercises 17-28 compute the determinant of the given matrix. A= (4) 18. B = (-4) 19. (~ ...

Linear Systems of First-Order Differential Equations 349 In exercises 37-40 show that the given set of vector functions are line ...

350 Ordinary Differential Equations or the equiva lent equation Ax - .Ax= (A - .AI)x = 0 if and only if (2) det (A - .AI) = 0. I ...

Linear Systems of First-Order Differential Equations 351 Then x 1 must satisfy -3 ) (Xu) -2 - 1 X21 = (1 -3) (Xu) = (0). 1 - 3 X ...

352 Ordinary Differential Equations x 12 and x22- Hence, x 12 = X22 and X22 is arbitrary. Choosing x22 = 1, we find an eigenvect ...

Linear Systems of First-Order Differential Equations Multiplying we find x 1 and x 2 must simultaneously satisfy XI+ X2 = 0 -X1 ...

354 Ordinary Differential Equations of multiplicity m = 1 and 2 is a n eigenvalue of the matrix A of multiplicity m=2. Let X1 = ...

Linear Systems of First-Order Differential Equations 355 Let be an eigenvector of the matrix A associated with the eigenvalue .A ...

356 Ordinary Differential Equations EXAMPLE 4 A 2 x 2 Matrix with Complex Eigenvalues and Eigenvectors Find the eigenvalues and ...

Linear Systems of First-Order Differential Equations 357 first equation.) Consequently, system (5) reduces to a single condition ...

358 Ordinary Differential Equations which accompanies this text contains a program named EIGEN, which com- putes the eigenvalues ...

Linear Systems of First-Order Differential Equations 359 by hand, we selected c = 1 and got a different eigenvector than the eig ...

360 Ordinary Differential Equations single eigenvector we calculated by hand in example 2 was x = ( -~). Notice that x 1 and x 2 ...

Linear Systems of First-Order Differential Equations 361 THE MATRIX WHOSE EIGE!VVALUES AND EIGENVECTORS ARE TO BE CALCULATED IS ...

362 Ordinary Differential Equations THE HA.TRIX WHOSE EIGENVALUES AND EIGENVECTORS ARE TO BE CALCULATED IS l .OOOOEtOO -2.0000Et ...