# If re i 0 for each i then the components of ut remain

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Unformatted text preview: om SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.4 Systems of Diﬀerential Equations http://www.amazon.com/exec/obidos/ASIN/0898714540 7.4 SYSTEMS OF DIFFERENTIAL EQUATIONS 541 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Systems of ﬁrst-order linear diﬀerential equations with constant coeﬃcients were used in §7.1 to motivate the introduction of eigenvalues and eigenvectors, but now we can delve a little deeper. For constants aij , the goal is to solve the following system for the unknown functions ui (t). u1 = a11 u1 + a12 u2 + · · · + a1n un , u2 = a21 u1 + a22 u2 + · · · + a2n un , . . . un = an1 u1 + an2 u2 + · · · + ann un , with u1 (0) = c1 , u2 (0) = c2 , . . . un (0) = cn . T H D E (7.4.1) Since the scalar exponential provides the unique solution to a single diﬀerential equation u (t) = αu(t) with u(0) = c as u(t) = eαt c, it’s only natural to try to use the matrix exponential in an analogous way to solve a system of diﬀerential equations. Begin by writing (7.4.1) in matrix form as u = Au, u(0) = c, where ⎛ u (t) ⎞ 1 ⎜ u2 (t) ⎟ u = ⎜ . ⎟, ⎝.⎠ . un (t) ⎛ IG ⎞ a1n a2n ⎟ . ⎟, .⎠ . a11 ⎜ a21 A=⎜ . ⎝. . a12 a22 . . . ··· ··· .. . an1 an2 ⎞ c1 ⎜ c2 ⎟ c = ⎜ . ⎟. ⎝.⎠ . · · · ann R Y and ⎛ cn If A is diagonalizable with σ (A) = {λ1 , λ2 , . . . , λk } , then (7.3.6) guarantees P eAt = eλ1 t G1 + eλ2 t G2 + · · · + eλk t Gk . O C (7.4.2) The following identities are derived from properties of the Gi ’s given on p. 517. k i=1 • deAt /dt = • AeAt = eAt A • e−At eAt = eAt e−At = I = e0 λi eλi t Gi = k i=1 λ i Gi k λi t Gi i=1 e = AeAt . (by a similar argument). (7.4.3) (7.4.4) (by a similar argument). (7.4.5) Equation (7.4.3) insures that u = eAt c is one solution to u = Au, u(0) = c. To see that u = eAt c is the only solution, suppose v(t) is another solution so that v = Av with v(0) = c. Diﬀerentiatin...
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## This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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