Calculus Solutions 32

Calculus Solutions 32 - Answers to Odd-Numbered Problems 3...

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Unformatted text preview: Answers to Odd-Numbered Problems 3 1 H 33 F if b 2 0; T; T; F (eAt is not a vector); T = [ -2 2 -: ] Section 16.2 Differential Equations (page 610) 13Best - Best = 8est gives B = 4 : y = 4est 3 y = 3 - 2 t + t 2 5 A e t + 4 e s t = 7 a t t = O i f A = 3 7 Add y = Ae-' because y' + y = 0; choose A = -1 so -e-' + 3 - 2t + t2 = 2 at t = 0 e" - 1 tekt g y = *;,= t; by19~bpital lim- = lim- = t k+O k k+O 1 11Substitute y = Aet + Btet + C cost + D sin t in equation: B = 1, C = i, D = - i, any A 13Particular solution y = Atet + Bet; y' = Atet + (A + B)et = c(Atet + Bet) + tet -1 g i v e s A = c A + l , A + B = c B , A = & , B = = 15X2eXt + 6XeXt + 5eXt = 0 gives X2 + 6X + 5 = 0, (A + 5)(X + 1) = 0, X = -1 or -5 (both negative so decay); y = Ae-' + Be-5t 1 7 (A2 + 2X + 3)eXt = 0, X = -1 f \/=Z has imaginary part and negative real part; + ~ ~ ( - 1 - f i ~ i ) t ; y = ~ ~ ( - l + f i i P y = Ce-' cos f i t + De-' sin f i t 19d = 0 no damping; d = 1underdamping; d = 2 critical damping; d = 3 overdamping...
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