MATH 2090  Review Problems for Exam 4  Fall 2000

1. For each of the matrices A below, find a fundamental set of solutions (explain why the solutions you find form a fundamental set), a fundamental matrix, and the general solution of the system x⃗′ = A x⃗. In each case, determine if the matrix A is defective or diagonalizable. If A is diagonalizable, find an invertible matrix S and a diagonal matrix D so that S^{−1}AS = D. Also, solve the initial value problem indicated in part (a).

(a) A = [ \begin{matrix}1 & 2 \\ 8 & 1 \end{matrix} ] , x⃗(0) = [ \begin{matrix}3 \\ 2 \end{matrix} ] (b) A = [ \begin{matrix}4 & −5 \\ 1 & 2 \end{matrix} ] (c) A = [ \begin{matrix}7 & −4 \\ 4 & −1 \end{matrix} ] (d) A = [ \begin{matrix}2 & −1 & 3 \\ 3 & 1 & 0 \\ 2 & −1 & 3 \end{matrix} ]

(e) A = [\begin{matrix}2 & −3 & 2 \\ −1 & 0 & 2 \\ −1 & −3 & 5 \end{matrix} ]

2. The differential equation x^{3}y′′′ + x^{2}y′′ − 2xy′ + 2y = 0, x > 0 has three solutions of the form y(x) = x^{r}. Find these solutions, show that they are linearly independent on the interval (0, ∞), and find the general solution of this differential equation.

3. For each of the non-homogeneous linear differential equations P (D)y = F (x) below, find the complementary solution, find the annihilator of the function F (x), and determine the form of a particular solution. In parts (a) and (b), also find the general solution of the differential equation, and solve the initial value problem in part (a).

(a) (D − 1)(D − 2)(D − 3)y = 6e^{4x}, y(0) = 4, y′(0) = 10, y′′(0) = 30 (b) y′′′ + 3y′′ + 3y′ + y = 2e^{−x} + 3e^{2x} (c) (D^{2} − 2D + 2)(D − 3)^{2}(D + 2)y = x^{2} − e^{x} cos x + 3e^{3x}

4. For each of the following functions, find the Laplace transform. In parts (a) and (b), use the definition to determine the Laplace transform of the function.

(a) f(t) = e^{−3t} (b) g(t) = sin 2t (c) h(t) = 2e^{−3t} − 3 sin 2t

5. Find the inverse Laplace transform of each of the following functions. (You may use the table on page 543.)

(a) F (s) = \frac{3}{s − 2} (b) G(s) = \frac{1}{s^{2} + 4} (c) H(s) = \frac{4}{s^{2}} − \frac{s + 2}{s^{2} + 9} (d) I(s) = \frac{1}{s(s − 1)}