hw6-solutions

1 cost sint therefore the functions ut et 2 cost

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Unformatted text preview: rts: x(t) = e−t (cos(t) + i sin(t)) 2+i 2 cos(t) − sin(t) cos(t) + 2 sin(t) = e−t + i e−t . 1 cos(t) sin(t) Therefore, the functions u(t) = e−t 2 cos(t) − sin(t) , cos(t) w(t) = e−t cos(t) + 2 sin(t) sin(t) form a fundamental system of solutions. In other words, the general solution of the differential equation y ′ (t) = Ay (t) is given by y (t) = a1 e−t 2 cos(t) − sin(t) cos(t) + 2 sin(t) + a2 e−t cos(t) sin(t) for suitable constants a1 , a2 . The initial condition y (0) = 1 1 implies a1 = 1 and a2 = −1. Thus, y (t) = e−t cos(t) − 3 sin(t) . cos(t) − sin(t) Problem 10: We consider the differential equation y ′ (t) = −3 2 y (t) −1 −1 with initial condition y (0) = 1 . −2 We begin by computing the eigenvalues of the coefficient matrix A: 0 = det λ + 3 −2 =...
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This note was uploaded on 02/09/2014 for the course MATH 53 taught by Professor Staff during the Winter '08 term at Stanford.

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