7. (5%) A semi-infinite plate coincides with the region defined by. The left end is held at temperature: exp(-y), and the right end is held at temperature zero for y 2 0. The bottom of the plate is insulated.
(A)g(a)=1+α,
(B)f(x)=αx,
(C)f(x)=α(ε-x),
(D) p(y) = sin(αy),
(E) p(y)=cos(ay).

複選題3. (5%) For the differential equation below.Which of the following are correct? (A)General solution form include (x)=x•cos(x), (B) General solution form include(C) Gencral solution formn include(D)(E)

複選題4. (5%) A heart pacemaker consists of a switch, a battery, a capacitor with constant capacitance C, and the heart as a resister with constant resistance R. When the switch is closed, the capacitor charges； when the switch is open, the capacitor discharges, sending an electrical stimulus to the heart. During the time the heart is being stimulated, the voltage y(t) across the heart satisties the linear differential equation : , where t is the time variable. Assume that initially, i.e, at t = O, the voltage y(t) across the heart is equal to 4 and the constant RC =1. Which of the following statements are true? (A) For t≥0, the particular solution of the lincar differential equation is (B) For t≥0,the particular solution of the linear differential equation is(C) For t≥0, the particular solution of the linear differential equation is, c is any constant, (D) For t≥0, the solution of the linear differential equation is(E) For t≥0,the solution of the linear diffcrential equation is

複選題5. (5%) Consider one specially designed circuit with one capacitor, one inductor, and four resisters. The output voltage of the capacitor is denoted as x(t) and the output current of the inductor is denoted as y(t). Assume that initially x(0)-0 and y(0)=0 and these two devices might interact with each other by the following behavior. For the capacitor, the changing rate of x(t) is declined at a rate of -3x(), and simultaneously increased at a rate of y(t), and also positively depends on the independent current source g(t) =3t. Similarly, for the inductor；, the changing rate of y(t) declines at a rate of -4y(t), and simultaneously increases at a rate of 2x(), and also positively depends on another independent voltage source. Which of the following statements are true?  (A) (B)(C)(D)(E)

6. (5%) A forced, undamped, and resonant motion of a mass on a spring can be described in the following equation: , where x(t) is the location of the mass. In the beginning, the mass is  located at x(0)=0 and the initial velocity of the mass is equal to I, ie.,， The Laplace transformX(s) of x(t) is: (A)(B)(C) (D)(E)

複選題8. (5%) A string is stretched and secured on the x-axis at x = 0 and x = 1 for t＞ 0, that is initially held at these points 0.01sin(3tx) and then simultaneously reicased at all points at time t = 0. The string is released from rest from the initial displacement. (Wave equation constant: a)u(x,t) = 0.01f(x)g(t) (A) g(t)=cos(πat),(B) f(x)=sin(3πx),(C) f(x)=sin(3παx), (D) g(t) = cos(3πat), (E) g(t) = sin(3πat).

9. (5%) An undamped string/mass system, in which the mass m = 1 slug and the spring constant k = 10 lb/t, is driven by the 2-periodic external force F(t) = 1-t,0 ＜t＜2； f(t+2) = f(t). Assume that when f(t) is extended to the negative t-axis in a periodic manner, the resulting function is odd. Find a particular solution Xp(t).(A) p(t)=nπt, (B)s=nπ, (C) k=nπ, (D) k=n2n2, (E) s=n2n2,

複選題10. (5%) Please pick the statements below that are true. (A) The rank of the coefficient mnatrix of a consistent system of linear equations is equal to the ariables in the genera number of basic varia ral solution of the system. (B) The matrix-vector product of an m X n matrix A and a' vector in Rn is a linear combination of the columns of A. (C) A subset of Rn containing fewer than n vectors must be linearly independent. (D) If S1 and S2 are finite subsets of R" having equal spans, then S1and S2 contain the same number of vectors. (E) If the columns of an n X n matrix A for a generating set of Rn, then the reduced row echelon form of A is In.

複選題11. (5%) Please pick the statements below that are true. (A) If A and B are m x n matrices and C is an n X p matrix, then (A + B)C = AC + BC. (B) For any matrices A and B, if A is the inverse of BT, then A is the transpose of B-I. (C) An n x n matrix is invertible if and only if its rows are linearly independent. (D) The codomain of any function is contained in its range. (E) for every standard vector e；, then f(v) mg(v) for every v in Rn.

複選題12. (5%) Please pick the statements below that are true. (A) A function is onto if its range equals its domain. (B) A linear transformation is one-to-one if and only if every vector in its range is the image of a unique vector in its domain. (C) For any m X n matrices A and B and any scalars c and d, (CA + dB)T = cAT + dBT. (D) No scaling operations are required in the forward pass of Gaussian elimination. (E) If A and B are invertible n X n matrices, then A + B is invertible.

13. (5%) Consider the matrix Which of the following values of c will make the matrix A invertible? (A) 3； (B) 4； (C) 5； (D) 6； (E) None of the above.