exercises7 - Systems of differential equations (the...

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Unformatted text preview: Systems of differential equations (the complex case) and double integrals over rectangles 1. Find the general solution of the system: (a) x1 = 2x1 − 5x2 x2 = 4x1 − 2x2 x1 = x1 − 2x2 x2 = 2x1 + x2 x1 = 5x1 − 9x2 x2 = 2x1 − x2 f (x, y ) dxdy : R (b) (c) 2. Evaluate (a) f (x, y ) = x2 + y 2 , R = {(x, y ) ∈ R2 : − 1 ≤ x ≤ 1, 0 ≤ y ≤ 1} or [−1, 1] × [0, 1] (b) f (x, y ) = x4 y + y 2 , R = {(x, y ) ∈ R2 : − 1 ≤ x ≤ 1, 0 ≤ y ≤ 1} or [−1, 1] × [0, 1] (c) f (x, y ) = y cos x + 2, R = {(x, y ) ∈ R2 : 0 ≤ x ≤ π/2, 0 ≤ y ≤ 1} or [0, π/2] × [0, 1] Answers: 1. (a) By considering the eigenvalue λ = 4i and its corresponding eigenvec1 +i , we obtain: tor v = 2 1 1 1 C cos(4t) − sin(4t) + C2 cos(4t) + sin(4t) X (t) = 1 2 2 C1 cos(4t) + C2 sin(4t) (b) By considering the eigenvalue λ = 1 + 2i and its corresponding eigeni vector v = , we obtain: 1 −C1 et sin(2t) + C2 et cos(2t) X (t) = C1 et cos(2t) + C2 et sin(2t) (c) By considering the eigenvalue λ = 2 + 3i and its corresponding eigen3 + 3i vector v = , we obtain: 2 3C1 e2t (cos(3t) − sin(3t)) + 3C2 e2t (cos(3t) + sin(3t)) X (t) = 2C1 e2t cos(3t) + 2C2 e2t sin(3t) 2. (a) 4/3 (b) 13/15 (c) π + 1 2 ...
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