2.[10%]是非題: (1) In any vector space,.
(A)O
(B)X

1.[10%] 多選題:Whichofthe following IVP has a nque solution? (A)(y-1)=6x,y(0)=1 (B)(y2+1)=x2,(3)=1 (C)y'=x,y(0)=1 (D) x2y"-4xy'+4y=0,y(3)=1,y'(3)=-1 (E) 3y"-4y'+1=ex,y(1)=0,y'(1)=1

(2) If V is a vector space other than the zero vector space, V contains a subspace W such that W≠V.(A)O(B)X

(3) Each element of a set S is a linear combination of the other elements of S if S is a linearly dependent set.(A)O(B)X

(4) A subset of a linearly independent set is also linearly independent(A)O(B)X

(5) The span of an empty set Φ is also Φ.(A)O(B)X

20、Consider the following partial differential equation for the bivariate function v(x, t)subject to conditionsExpress the solution v(x, t) in the following formfor some constants an, bn, pn, qn, rn ∈ R satisfying 0 ≤ r0 ＜ r1 ＜ r2 ＜ ... and ＞0for n = 1, 2, ... Which of the following statements is/are true?(A) a0 = - (B) b1 = (C) p3 = (D) r4 = (E) None of the above is true.

19、Continue from Question 十八. Which of the following statements is/are true regarding the solution y(t)?(A) y'(1) = 1 (B) y'(π) = 1 (C) y"(1) = -1 (D) y"(π) = 0 (E) None of the above is true.

18、Consider the following differential equation for the function y(t) defined for t ≥ 0where u(t) is the usual unit-step (Heaviside) function and where * is the usual convolutional operation of two functions. Assume y(t) = 0 for t ＜ 0. Given the initial condition y(0) = 1, which of the following statements is/are true regarding the Laplace transform Y(s) = {y(t)} of solution y(t) in the above equation?(A) Y(s =) = −6(B) Y(s = 1) = 1(C) Y(s = 2) = (D) Y(s = 3) = (E) None of the above is true.

17、Continue from Question 十六. Assuming b₂ = 2, which of the following statements is/are true regarding the solution y₂(t)?(A) (B) b₁ = 0 (C) (D) (E) None of the above is true.

16、Consider the following second order differential equation3t2y''(t) + sin(t)y'(t) - cos(t)y(t) = 0.Let y1(t) = tr1be two linearly independent Frobenius series solution for y(t) in the above differential equation when t ＞ 0. Assuming r1≥r2 and a0 = 1, which of the following statements is/are true regarding the solution y1(t)?(A) r1 = 1. (B) (C) (D) (E) None of the above is true.

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