十四、 The first order system

can be reduced into an equivalent second order system y"(t) = By(t) with y(t) =
. Which of the following statements is/are true?
(A)
(B)
(C) The eigenvalues of B are 9 and 25.
(D) is an eigenvector for B.
(E) None of the above are true.

複選題十、Let , be a linear transformation on for a rotation by an angle θ about a unit vectoru. Specifically, we let the matrix for with respect to the standard basis S for bewhere is the coordinate vector of u with respect to S, c = cos(θ)and s = sin(θ). Furthermore, let A be the rotation matrix about the z-axis of Cartesiancoordinate system by an angle θ, i.e.,where be an ordered orthonormal basis for with and let be the changeof-basis matrix for changing basis from B to S. Which of the following statements is/aretrue?(A), n1n2 +b1b2 + u1u2 = 0 and n1n3 + b1b3 +u1u3 =0.(B) The coordinate vector of u with respect to basis B is .(C) (D) (E) The matrix for with respect to basis B is A.

複選題十一、 Solve for y(x) the first order differential equationxy'(x) - 4x2y(x) + 2y(x)ln(y(x)) = 0by the substitution u = In(y(x)). Which of the following statements is/are true?(A) It is a nonlinear ordinary differential equation for the dependent variable y.(B) It is a nonlinear ordinary differential equation for the new variable u.(C) There exists a solution y(x) satisfying the condition y(0) = 1.(D) There exists a solution y(x) satisfying the condition y(1) = 1.(E) None of the above are true.

複選題十二 、 The Cauchy-Euler equationx2y"(x)-4xy'(x)+6y(x)=0can be transformed into a constant coeficient equation y"(u) + by'(u) + cy(u) = 0 by thesubstitution u = In(x). Which of the following statements is/are true?(A) b= -5.(B) c=-6.(C) The solution y(x) exists only for x ＞ 0.(D) y(x) = C1x2 + C2x3 for some constants C1 and C2.(E) None of the above are true.

十三、Continued from Problem 十二 . Solve the non-homogeneous second order differential equa-tionx2y"(x)-4xy'(x)+6y(x)=x3by variation of parameters, i.e, set the particular solution as +u2(x)y2(x), where y1(x) and y2(x) are homogeneous solutions. With initial conditionsy(1) = 0 and y'(1) = 1 for the complete solution y(x), which of the following statementsis/are true?(A) The real valued solution y(x) exists only for t ＞ 0.(B) y(2) = 2.(C) 2y'(2) -3y(2) = 8.(D) y'(3) -y(3) = 4.(E) None of the above are true.

十五、Continued from Problem +u. Find the partieular solution for the second order system y"(t) = By(t) with initial conditions . Which of the following statements is/are true? (A) (B) (C) (x1(t) +x3(t)) is an odd function in t.(D) (x1(t) -x3(t)) is an odd function in t.(E) None of the above are true .

十六 、For s ＞ 0, let F(s) be the unilateral Laplace transform of function f(t) given byWhich of the following statements regarding the values of f(t) is/are true?(A) (B) f(2) = 1.(C) f(4)=2.(D) f(8) = 3.(E) None of the above are true.

十七 、 Consider the following periodic functionwhich has a Fourier series representationfor some . Which of the following statements is/are true?(A) (B) (C) (D) (E) None of the above are true.m＞o

十八、 For the following second order differential equationtx"(t)+(4t-2)x'(t)+(2t-4)x(t)=0Let x1(t)= be the two linearly independentFrobenius series solutions for x(t) when x ＞ 0, where ri and r2 are the zeros of thecorresponding indicial equation. Assume r1 ≥ r2 and ao = b0 = 1. Which of the followingstatements is/are true?(A) r1 - r2 is not an integer.(B) .(C) (D) .(E) None of the above are true.

十九、Continued from Problem 十i. The second order differential equation can be alternativelysolved by using Laplace transform. Assuming x(0) = 0 and fo = 1, which ofthe following staterents is/are true about the values of x(t) and its unilateral Laplacetransform X(s) = ?(A) x'(0) = 1.(B)x(1)=1.(C) X(1) ＜1.(D) Values of X(s) exists for all s ＞ -1.(E) None of the above are true.

二十、Consider the following boundary value problem for the bivariate function u(x,t) definedfor 0 ≤x≤π and t≥0Given the end-point and initial conditions=0 and u(x,0)=x(π-x)which of the following statements is/are true for the solution u(x,t) when it is expressed as for some constants (A) (B) a1=-1.(C) p2 =-31.(D) b1 =-1(E) None of the above are true.