MATH 2090  Review Problems for Exam 4  Fall 2001

1. For each of the matrices A below, find a real-valued fundamental set of solutions (explain why your solutions form a fundamental set), a fundamental matrix, and the general solution of the system x⃗′ = A x⃗. Determine if A is defective or diagonalizable in each case. If A is diagonalizable, find a diagonal matrix D and an invertible matrix S 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. Suppose λ is an eigenvalue of the matrix A, and that v⃗ is a corresponding eigenvector.

(a) Show that λ is an eigenvalue of the matrix A^{T}.

(b) Show that λ^{2} is an eigenvalue of the matrix A^{2}, and that v⃗ is a corresponding eigenvector.

(c) If A is non-singular, show that 1/λ is an eigenvalue of A^{−1}, and that v⃗ is a corresponding eigenvector.

5. Let X(t) = [ x⃗_{1}(t) x⃗_{2}(t) ⋯ x⃗_{n}(t) ] be a fundamental matrix for the system x⃗′ = A(t) x⃗ on the interval I.

(a) Show that X′(t) = A(t) X(t).

(b) Show that the general solution of the system can be written as x⃗(t) = X(t) c⃗, where c⃗ is a constant vector.

(c) If t_{0} is in I, show that the solution of the IVP x⃗′ = A(t) x⃗, x⃗(t_{0}) = α⃗ can be written as x⃗(t) = X(t)X^{−1}(t_{0})α⃗.