Practice 4 Solutions

Equation y ay where 200 a 2 2 1 221 2 y 0

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Unformatted text preview: eAt where A is the matrix 221 A = 0 2 1 012 0 −1 1 033 1 −1 −2 3t e e 0 1 −1 + e2t 0 0 1 1 0 0 + = 2 2 0 −1 1 011 0 0 0 t λ = 1, 2, 3 eAt [8] Find eAt where A is the matrix 120 A = 1 0 1 112 1 −3 1 1 1 −1 111 t 3t e e e −1 3 −1 + 0 0 0 + 1 1 1 = 4 2 4 0 0 0 −1 −1 1 222 −t λ = −1, 1, 3 eAt , 3 × 3 Exercise Set C (distinct roots), November 3 × 3 Exercise Set C (distinct roots) Linear Algebra, Dave Bayer, November , [1] Solve the di erential equation y = Ay where 120 A = 1 1 2 , 011 eAt 0 y ( 0) = 1 1 2 −4 4 2 0 −4 244 t 3t e e e −2 4 −4 + 00 0 + 2 4 4 = 8 4 8 1 −2 2 −1 0 2 122 −t λ = −1, 1, 3 0 −4 8 t 3t e e + e 8 0+ 0 y= 8 4 8 0 2 4 −t [2] Solve the di erential equation y = Ay where 222 A = 0 2 0 , 121 2 y ( 0) = 0 1 λ = 0, 2, 3 eAt 1 1 −2 282 0 −3 0 3t e 1 0 0 + e 2t 0 0 0 0 1 0 + = 0 3 3 −1 −1 2 141 0 −1 0 0 6 0 3t 1 2t + e 0 0 +e 0 y= 3 3 0 3 0 [3] Solve the di erential equation y = Ay where 210 A = 2 2 1 , 022 2 y ( 0) = 0 1 λ = 0, 2, 4 eAt 2 −2 1 2 0 −1 221 2t 4t 1 e e 4 −2 + 00 0 + 4 4 2 = −4 8 4 8 4 −4 2 −4 0 2 442 5 3 5 2t 4t 1 e e 0 + 10 y = −10 + 8 4 8 10 −6 10 , 3 × 3 Exercise Set C (distinct roots), November [4] Solve the di erential equation y = Ay where 200 A = 2 2 1 , 221 2 y ( 0) = 0 1 λ = 0, 2, 3 eAt 0 0 0 000 100 3t 1 e 1 −1 + e2t −2 0 0 + 6 2 1 = 0 3 3 0 −2 2 621 −2 0 0 0 0 2 3t e 1 13 y = −1 + e2t −4 + 3 3 2 13 −4 [5] Solve the di erential equation y =...
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