複選題
十二 、 The Cauchy-Euler equation
x²y"(x)-4xy'(x)+6y(x)=0
can be transformed into a constant coeficient equation y"(u) + by'(u) + cy(u) = 0 by the
substitution 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.

八、 Consider a 3 x 3 nonzero matrix A, and let W = . The leastsquares solutions xLs to Ax = b are illustrated in the following figure，where projwe b is theorthogonal projection of vector b onto vector space W. A minimun length least squares, solution EMurs is the one among the least squares solutions that has a minimum norm.Which of the following stateme ents is/are true?(A)(B) (C) (D) The minimum length least squares solution EMLLS to Ax = b is unique, and =projvxLs, where V = .(E) None of the above are true.

九 、 Let v1 and v2 be the eigenvectors of matrix A corresponding respectively to eigenvalues λ1 and λ2, where  It is found that λ1 > 1 > |λ2l. With matrix A, ,be a series of vectors related by . Given the initial condition x0 = = , which of the following statements is/are true? (A)(B)(C) (D) (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.

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.

十四、 The first order systemcan 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.

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.