1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 363 8.3 Linear Systems with Constant Coefficients Numerous physical phenome ...

364 Ordinary Differential Equations Letting uI = YI (the position of pendulum 1), u2 = y~ (the velocity of pendulum 1), u3 = Y2 ...

Linear Systems of First-Order Differential Equations 365 If y is an n x 1 column vector whose entries are functions of a single ...

366 Ordinary Differential Equations subject to the n constraints (4b) If we let y(x), b(x), and d be then x 1 column vectors Y2( ...

Linear Systems of First-Order Differential Equations 367 The development of the theory for systems of linear first-order differe ...

368 Ordinary Differential Equations DEFINITION General Solution of a Homogeneous Linear System When Y1, y2, ... , Yn are linearl ...

Linear Systems of First-Order Differential Equations 369 are linearly independent solutions of the homogeneous linear system (8) ...

370 Ordinary Differential Equations To determine whether the set {y 1 (x),y2(x)} is linearly dependent or lin- early independent ...

Linear Systems of First-Order Differential Equations REPRESENTATION THEOREM FOR NONHOMOGENEOUS LINEAR SYSTEMS 371 If A ( x) is c ...

372 Ordinary Differential Equations SOLUTION Differentiating Yp, we find Yp / = (-1) 2x. Computing A(x)yp + b(x), yields = (-2-1 ...

Linear Systems of First-Order Differential Equations 373 Cancelling the nonzero scalar factor erx, we find the unknowns v and r ...

374 Ordinary Differential Equations THE MATRIX WHOSE EIGENVALUES AND EIGENVECTORS ARE TO BE CALCULATED IS 2.0000EtOO O.OOOOEtOO ...

Linear Systems of First-Order Differential Equations 375 ( .707107) (-.707107) (0) y(x) = C1e^3 x 0.000000 + c2ex 0.000000 + c3e ...

376 Ordinary Differential Equations A THEOREM ON OBTAINING REAL SOLUTIONS FROM COMPLEX CONJUGATE EIGENVALUES AND EIGENVECTORS If ...

Linear Systems of First-Order Differential Equations 377 an associated eigenvector, ).. and x satisfy Ax = >.x. Taking the co ...

378 Ordinary Differential Equations and Y2 ( x) = e^2 x sin x ( -~) + e^2 x cos x ( -~) _ (-e^2 x sin x -e^2 x cos x) e^2 x cos ...

Linear Systems of First-Order Differential Equations 379 The eigenvalues are 1+2i = o+{3i, 1-2i, and 2 and the associated eigenv ...

380 Ordinary Differential Equations Consider the homogeneous linear system (24) a. Verify that are linearly independent soluti ...

Linear Systems of First-Order Differential Equations 381 C2 -1 -:) G 1 D -1 -2 12.^1 1 -1 -2 2 (-! 1 0 -~) (-! 1 0 13. (^2 ...

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