Fall 2000  Math 2090  Exam 4

1. Determine if the matrix A = [\begin{matrix}2 & 0 & 1 \\ 0 & 3 & 0 \\ −1 & 0 & 4 \end{matrix} ] is defective or non-defective.

det(A − λI) = (3 − λ)^{3} , so λ = 3 is an eigenvalue with algebraic multiplicity 3.

ker(A − 3I) = span{(1, 0, 1), (0, 1, 0)}, so the geometric multiplicity of λ = 3 is 2 < 3, and A is defective.

2. Find all eigenvalues and associated eigenvectors of the matrix A = [ \begin{matrix}1 & −4 \\ 1 & 1 \end{matrix} ].

det(A − λI) = λ^{2} − 2λ + 5, so λ_{1} = 1 + 2 i and λ_{2} = 1 − 2 i. Associated eigenvectors are v⃗_{1} = (−2 i, 1) and v⃗_{2} = (2 i, 1)

3. Consider the differential equation (D − 2)(D^{2} − 2D + 5)y = 3e^{2x} + 7x.

(a) Find the general solution of the associated homogeneous differential equation,

(D − 2)(D^{2} − 2D + 5)y = 0.  y(x) = c_{1}e^{2x} + c_{2}e^{x} cos(2x) + c_{3} sin(2x)

(b) Find the annihilator of the function F (x) = 3e^{2x} + 7x. D^{2}(D − 2) annihilates F (x) = 3e^{2x} + 7x.

(c) Find the form of a particular solution of the non-homogeneous differential equation given above.

y_{p}(x) = Axe^{2x} + Bx + C

4. (a) Use the definition to compute the Laplace transform of the function f(t) = e^{6t}.

L[e^{6t}] = ∫_{0}^{∞} e^{−st}e^{6t} dt = ∫_{0}^{∞} e^{(6−s)t} dt = lim_{N→∞} [ \frac{e^{(6−s)t}}{6 − s} |_{0}^{N}]

= lim_{N→∞} [\frac{e^{(6−s)N}}{6 − s} − \frac{1}{6 − s}] = −\frac{1}{6 − s} = \frac{1}{s − 6} if 6 − s < 0

(b) Find the Laplace transform of the function f(t) = 4t^{3} − 2 sin 3t.  L[f(t)] = \frac{24}{s^{4}} − \frac{6}{s^{2} + 9}

(c) Find the inverse Laplace transform of the function F (s) = \frac{4}{s^{3}} + \frac{3}{s + 4} + \frac{2s}{s^{2} + 4}.

L^{−1}[F (s)] = 2t^{2} + 3e^{−4t} + 2 cos(2t)

5. In each part below, find a fundamental set of solutions for the system of differential equations x⃗′ = Ax⃗. Your answers should be real-valued.

(a) A = [\begin{matrix}3 & −5 \\ 2 & 1 \end{matrix}] an eigenvalue/eigenvector pair is λ_{1} = 2 + 3 i, v⃗_{1} = [\begin{matrix}1 + 3 i \\ 2 \end{matrix}].

x⃗_{1}(t) = e^{2t} cos(3t) [\begin{matrix}1 \\ 2 \end{matrix}] − e^{2t} sin(3t) [\begin{matrix}3 \\ 0 \end{matrix}], x⃗_{2}(t) = e^{2t} sin(3t) [\begin{matrix}1 \\ 2 \end{matrix}] + e^{2t} cos(3t) [\begin{matrix}3 \\ 0 \end{matrix}]

(b) A = [\begin{matrix}13 & −9 \\ 4 & 1 \end{matrix}] (defective); eigenvalues: λ_{1} = λ_{2} = 7; associated eigenvector: v⃗ = [\begin{matrix}3 \\ 2 \end{matrix}].

x⃗_{1}(t) = e^{7t} [\begin{matrix}3 \\ 2 \end{matrix}], x⃗_{2}(t) = te^{7t} [\begin{matrix}3 \\ 2 \end{matrix}] + e^{7t} [\begin{matrix}2 \\ 1 \end{matrix}] is one fundamental set

6. The eigenvalues of certain 2 × 2 matrix A are λ_{1} = 3 and λ_{2} = 2.

Eigenvectors corresponding to λ_{1} and λ_{2} are v⃗_{1} = [ \begin{matrix}1 \\ 2 \end{matrix} ] and v⃗_{2} = [ \begin{matrix}1 \\ 3 \end{matrix} ].

(a) Solve the initial value problem x⃗′ = Ax⃗, x⃗(0) = [ \begin{matrix}2 \\ 7 \end{matrix} ].  x⃗(t) = −e^{3t} [\begin{matrix}1 \\ 2 \end{matrix}] + 3e^{2t} [\begin{matrix}1 \\ 3 \end{matrix}]

(b) Find the matrix A.  A = SDS^{−1} = [\begin{matrix}1 & 1 \\ 2 & 3 \end{matrix}] [\begin{matrix}3 & 0 \\ 0 & 2 \end{matrix}] [\begin{matrix}3 & −1 \\ −2 & 1 \end{matrix}] = [\begin{matrix}5 & −1 \\ 6 & 0 \end{matrix}]