� � � � � � � � � 18.034 PRACTICE EXAM 3, SPRING 2004 Problem 1 Let A be a 2 × 2 real matrix and consider the linear system of first order differential equations, � � y�(t) = Ay(t), y(t) = y1(t) . y2(t) Let α, β be fixed real numbers, and let M1, M2 be fixed 2 × 2 matrices with real entries. Suppose that the general solution of the linear system is, y(t) = (k1M1 + k2M2) eαt cos(βt) , eαt sin(βt) where k1, k2 are arbitrary real numbers. (a) Prove that M1 and M2 each satisfy the following equation, AMi = MiD, D = α −β.β α (b) Consider the linear system of differential equations, z�(t) = A2z(t), z(t) = z1(t) . z2(t) Use (a) to show that for every pair of real numbers k1, k2, the following function is a solution of the linear system, � � e(α2−β2)t cos(2αβt)z(t) = (k1M1 + k2M2) e(α2. −β2)t sin(2αβt) Problem 2 Consider the following inhomogeneous 2nd order linear differential equation, ⎧ ⎨ y�� − y = 1, y(0) = y0,⎩ y�(0) = v0 Denote by Y (s) the Laplace transform, Y (s) = L[y(t)] = ∞ e−st y(t)dt. 0 (a) Find an expression for Y (s) as a sum of ratios of polynomials in s. (b) Determine the partial fraction expansion of Y (s). (c) Determine y(t) by computing the inverse Laplace transform of Y (s). Problem 3 The general skew-symmetric real 2 × 2 matrix is, 0 b A = −b,0 where b is a real number. Prove that the eigenvalues of A of the form λ = ±iµ for s ome real number µ. Find all values of b such that there is a single repeated eigenvalue. Date : Spring 2004. 1�� � Problem 4 Let λ be a real number and let A be the following 3 × 3 matrix,⎤⎡ λ 1 0 0 λ 1 A = ⎣ ⎦ . 0 0 λ Let a1, a2, a3 be real numbers. Consider the following initial value problem, (t) Ay(t),⎤⎡ ⎧ ⎪⎪⎨y = a1 ⎪⎪⎩ y(0) = ⎣ ⎦ a2 a3 Denote by Y(s) the Laplace transform of y(t), i.e., Y1(s) Y2(s) ⎤⎡ ⎣ ⎦ Y(s) Yi(s) = L[yi(t)], i = 1, 2, 3.= , Y3(s) (a) Express both L[y�(t)] and L[Ay(t)] in terms of Y(s). (b) Using part (a), find an equation that Y(s) satisfies, and iteratively solve the equation for Y3(s), Y2(s) and Y1(s), in that order. (c) Determine y(t) by applying the inverse Laplace transform to Y1(s), Y2(s) and Y3(s). Problem 5 For each of the following matrices A, c ompute the following, (i) Trace(A), (ii) det(A), (iii) the characteristic polynomial pA(λ) = det(λI − A), (iv) the eigenvalues of A (both real and complex), and (v) for each eigenvalue λ a basis for the space of λ-eigenvectors. (a) The 2 × 2 matrix with real entries, 0 1 A = −1 0 . Hint: See Problem 3. (b) The 3 × 3 matrix with real entries, ⎤⎡ 3 1 1 0 5 0 ⎣ ⎦ A = . 0 0 3
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