六、(10%,計算題)

 Find the solution y(x) of the following initial value problem for x ＞ 1. 

x²y"-2xy'+2y=xlnx, y(1)=1, y'(1)=0

(D).(3%) Use the set {1, x, x2, x3} as the basis for P3. Find the matrix representation of the linear transformation that does differentiation on polynomials in P3.

(E). (3%) Continue with part (D). Can the matrix representing differentiation be diagonalized? Why?

(A).(5%) (x + 3)2y"' -8(x+ 3)y'+ 14y =0

(B).(5%) x2y" - 4xy'+ 6y = Inx2

(A). (5%) Consider a system of linear differential equations in the following vector form: Here v(t) is an n-dimensional column vector and M is an n x n matrix. Assume that the matrix M can be diagonalized by a similarity transformation:Here S is the transformation matrix and D is a diagonal matrix with λ1,λ2,... ,λn as its diagonal elements. The initial values of the equations form the vector v(0) and P(t) is a diagonal matrix with exp (λ1t),exp(λ2t), ,exp (λnt) as its diagonal elements. Please express the solution v(t) for t＞ 0 in terms of v(0), S and P(t) and explain why.

(B). (5%) Continued from (A), if the matrix M is given by write down a transformation matrix S that can diagonalize M into D and also write down the corresponding diagonal matrix D.

(C). (5%) Consider a system of differential equations in the following vector form: Here the additional term f(t) is a known n-dimensional t-dependent column vector and the other symbols are the same as in (A). Please express the solution v(t) for t ＞ O in terms ofv(0), S, P(t), f(t) and explain why.

八、(5%,計算題) Using the Laplace transform metbod to find the solution y(x) for the following initial value problem: y'''(x)+4y'''(x)+11y"(x)+14y'(x)+10y(x)=0, y'''(0)=1,y"(0)=0, y'(0)=0,y(0) =0 (Hint:s4+4s3+11s2+14s+10=((s+1)2+1)((s+1)2+4)]

(A) (5%) can be expanded into a sine series as follows:Please determine the solution y(x) in the range of 0 ≤ x≤1 interms of Cm , m=1,2,...∞.

(B) (5%) Please determine the value of c1.