hw6-solutions

# Consequently the function xt e2it 1i 1 is a solution

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Unformatted text preview: (λ + 3)(λ + 1) + 2 = λ2 + 4λ + 5. 1 λ+1 Therefore, the eigenvalues are −2 + i and −2 − i. The vector v= 1−i 1 is an eigenvector of A with eigenvalue i. Consequently, the function x(t) = e(−2+i)t 1−i 1 is a solution of the diﬀerential equation x′ (t) = Ax(t). We now split x(t) into its real and imaginary parts: x(t) = e−2t (cos(t) + i sin(t)) = e−2t 1−i 1 cos(t) + sin(t) sin(t) − cos(t) + i e−2t . cos(t) sin(t) Therefore, the functions u(t) = e−2t cos(t) + sin(t) , cos(t) w(t) = e−2t sin(t) − cos(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−2t cos(t) + sin(t) sin(t) − cos(t) + a2 e−2t cos(t) sin(t) for suitable constants a1 , a2 . The initial condition y (0) = 1 −2 implies a...
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