14. The following second order system

can be transformed into an equivalent first order system

for some real matrix A by introducing the four functions x₁(t)= x(t),x₂(t)=x'(t),
x₃(t) = y(t) and x₄(t) = y'(t). Which of the following statements is/are true?
(A)
(B)
(C) -36 is one of the eigenvalues of A.
(D) 8 is one of the eigenvalues of A.

(E) None of the above is true.

10. Let P ∈ be the orthogonal projection matrix, with respect to the Euclidean innerproduct, of vectors in onto col( A ) for some nonzero matrix A ∈ . Which of thefollowing statements is/are true?(A) ≥rank(P).(B) P can have an eigenvalue larger than 2.(C) Px≠ x for all x e .(D) If k ＜ n and ! e , the system of linear equations Ax = o has a unique least squaressolution.(E) None of the above is true.

11.Consider the first order differential equation xy'(x) = y(x) - with initialcondition y(2) = 0. Which of the following statements is/are true?(A) It is a linear differential equation for the dependent variable y.(B) y(x) is a parabolic function of x.(C) y'(x) =-x.(D) y"(x) =.(E) None of the above is true.

12. For the following homogeneous second order linear differential equation(x2-1)y"(x)-2xy''(x)+2y(x)=0for x ＞ 1, given one solution y1(x) = x, the other linearly independent solution y2(x) canthen be derived by setting y2(x) = v(x)y1(x). Which of the following statements is/aretrue?(A)(x3+x)v"(x)+2v'(x)=0.(B) u(x) with u'(x) = corresponds to one of the possible solutions.(C) u(x) with u'(x) = corresponds to one of the possible solutions.(D) y(x) = 1 + x + x2 is one of the possible solutions.(E) None of the above is true.

13. Continue from Question 十=. Solve the following non-homogeneous second order lineardifferential equation(x2-1)y"(x)-2xy'(x)+2y(x)=x2-1for x ＞ I using the variation of parameters, i.e,, set the particular solution aswhere y1(x) and y2(x) are homogeneous solutions from Question 十二. Which of thefollowing statements is/are true?(A)(B)(C) (D)(E) None of the above is true.

15. Continue from Question +e. Find the particular solution of the second order system inequation (2) with initial conditions z(0) = 2, x'(0) = 12, y(0) = 1 and y'(0) = 6. Whichof the following statements is/are true?(A) .(B) (t)=2cos(6t) +2sin(6t).(C).(D) y(t) = cos(6t) + sin(6t).(E) None of the above is true.

16. Consider the following second order differential equation3t2y"(t) + sin(t)y'(t) - cos(t)w(t) =0. be two linearly independent Frobeniusseries solution for y(t) in the above differential equation when t ＞ 0. Assuming r1≥ r2and 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.

17.Continue from Question 十六.Assuming b0 = 2, which of the following statements is/aretrue regarding the solution y2(t)?(A) (B) b1=0(C)(D)(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 convolutionaloperation of two functions. Assume y(t) = 0 for t ＜ 0. Given the initial conditiony(0) = 1, which of the following staterents is/are true regarding the Laplace transformY(s) = ((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.

19. Continue from Question t十八. Which of the following statements is/are true regardingthe 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.

20、Consider the following partial differential equation for the bivariate function v(x, t), 0 ≤ x ≤ 10, t ≥ 0subject to conditionsExpress the solution v(x, t) in the following form for some constants ,  ∈ satisfying 0 ≤ r0＜r1＞r2＜... ＞ 0 for n = 1, 2,... Which of the following statements is/are true?(A)(B)(C) (D) (E) None of the above is true.