3. (5%) Which are the possible solutions of the differential equation y" - 4y'+ 4y = e^t?
(A) y = e^t
(B) y = e^2t
(C) y = e^t + e^2t
(D) y = te^t + e^2t

(E) y = e^t + te^2t

34. Which of the following arguments counters the argument made in this passage?(A) The more choices we have, the more freedom.(B) The more freedom we have, the less security.(C) The less freedom we have, the less choices.(D) The more security we have, the more freedom.

35. Which of the following examples best supports the argument presented in the passage?(A) If we provide more tenure positions for university teachers, they will have more space to decide for themselves which research area they want to focus on.(B) If we sign a trade treaty that helps open up a free market, we will be less able to keep inflation below two percent.(C) If we establish laws to regulate the right and obligation in marriage, we will have less freedom to build the kind of marriage relationships we want.(D) If a company doesn't set up some SOP and check whether the employees follow them, it is more likely to lose its competitiveness.

1. (5%) Which of the following is the possible integrating factor for y' -y =x4?(A) e-2(B) x-2(C) ln(x-2)(D) e2x-2(E) e-2x

2. (5%) Find the inverse Laplace transform of the function F(s) =.(A) cos3x + sin3x(B) cosh3x + sinh3x(C) e3x cos2x + e3xsin2x(D) e3x cosh2x + e3xsinh2x(E) e3x + e-3x

4. (5%) The mass-spring system is described by x" +bx' + 4x = 0, x(0)=4, x'(0)= 0.Under which condition, the system is in the state of underdamped motion?(A)b＞4(B)0＜b＜4(C)b = 4(D)b = 2(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)

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