Unformatted text preview: That is, ﬁnd a and b in the equation E ( t ) ≈ a + bt valid for small t . (e) (2 pts) Find one initial condition for which dE dt ² ² ² t =0 > 0, and one initial condition for which dE dt ² ² ² t =0 < 0. 2. (10 pts total) (a) (5 pts) Find the eigenvalues of a real 2 × 2 matrix A in terms of its trace and its determinant. Hence, ﬁnd the regions in the (Tr A, det A ) plane in which A has two negative real eigenvalues, two positive real eigenvalues, a pair of complex conjugate eigenvalues with negative real parts, etc. Also, ﬁnd the curve on which A has multiple eigenvalues. (b) (5 pts) For the planar diﬀerential equation ˙ x = Ax , x ∈ R 2 , sketch the phase portrait for each of the regions identiﬁed in (a). 1...
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- Fall '08
- Linear Algebra, pts, Orthogonal matrix, Applied Dynamical Systems