9. (5%) The differential equation x" + 3x' - 5x = sin t is equivalent to the system:
(A)x' = 3x - 5y； y' = sin t
(B) x' = 5x - 3y； y' = sin t
(C) x' = y； y' = 5x - 3y + sin t
(D)x' = y； y' = 3x - 5y + sin t

(E) None of above

5. (5%) Solve the equation y(5) - y(4) – 2y (3) + 2y" + y' = y. Which are the possible solutions?(A)y = C₁ex + c2xex+C3x²ex + C4x3ex+c5xe-x(B)y = C₁ex + C2xex+C3e-x + C4xe-x+c5x2e-x(C)y = C₁ex + C2xex+C3e-x + C4e-2x+C5xe-2x(D)y = c₁ex + c2xex+C3x²ex + C4e-x+c5xe-x(E) y = C₁ex + C2xex+C3ex + C4e2x+C5e-2x

複選題6. (5%) Which boundary value problems have nontrivial solutions? (Trivial solution is defined as y(x) = 0.)(A) y" + 2y' - 3y = 0； y(0) = 0, y'(1) = 0(B) y" = 0； y(-1) = 0, y(1) – 2y'(1) = 0(C) y" + y = 0； y(0) = 0, y(π) = 0(D) y" + y = 0； y(0) = 0, y(π) = 1(E) All of above

複選題7. (5%) Which functions satisfy the wave equation?(A)u(x,t) = (x² + k²)t²(B) u(x, t) = sin x cos kt(C) u(x, t) = tan⁻¹(x + kt)(D)u(x, t) = ln(x + kt)(E) u(x, t) = e⁻ᵏᵗ sin x

8. (5%) The initial value problem is described as follows: y" + 4y = 0； y(0) = 2, y'(0) = 2. The Laplace transform Y(s) of the solution y(t) is:(A)(B)(C) (D)(E)

複選題10. (5%) The differential equation is y' = in the standard form, and 2xydx + x²dy = 0 in a differential form. This differential equation is(A) linear(B) homogenous(C) exact(D) separable(E) Bernoulli

複選題11. (10%) Which of the following are correct?(A) The system defined by F(x,y)=(x², x) is linear.(B) The system defined by F(x,y)=(dx/dt, x) is linear.(C) A and B are m x n matrices. If Aw = Bw for all w in Rn, then A = B.(D) The columns of matrix A contains zero vector. If Ax=b have solution, it will have infinite solutions.(E) The zero vector of Rn is within the span of any finite subset of Rn.

複選題12. (10%) For the matrix A, which of the following are correct?A =(A) The rank of A is 4.(B) The rank of A is 3.(C) The dimension of null space of A is 0.(D) The dimension of null space of A is 1.(E) The dimension of null space of A is 2.

複選題13. (5%) Which of the following are correct?(A) Below vector set S is dependent.S =(B) Invertible matrix must be square matrix.(C) Square matrix must be invertible.(D) The inverse of matrix(E) A n x n matrix is invertible if its columns span Rn.

複選題14. (10%) For the matrix below, which of the following are correct?(A) -1 is its eigenvalue.(B) 1 is its eigenvalue.(C) -3 is its eigenvalue.(D) Zero vector belongs to its eigenvector.(E) [1 0 2]T is a basis of its eigenspace.

複選題15. (5%) Which of the following are correct?(A) An eigenvalue of a matrix has infinite number of eigenvectors.(B) If det(A – λIn) = 0, λ is the eigenvalue of n x n matrix A.(C) If λ is the eigenvalue of matrix A, the columns of (A -λIn) are independent.(D) If λ is the eigenvalue of matrix A, there exist a non-zero vector v such that Av = λv.(E) The number of the eigenvalue of n x n matrix A may larger than n.