課程名稱︰工程數學上 課程性質︰必修 課程教師︰蕭浩明 開課學院：工學院 開課系所︰機械系 考試日期（年月日）︰2017/1/9 考試時限（分鐘）：100 試題 : 1. (15%) Use Taylor series method to solve (x^2 + 2x + 1)y" - 2(x + 1)y' + 2y = 0 y(0) = a0, y'(0) = a1 2. (20%) Find the recurrence formula and general solution near t = 0 for the differential equation y" + (t - 1)y' + (2t - 3)y = 0 3. (20%) Solve X' = 6x + y + 6t Y' = 4x + 3y - 10t + 4 by using the Undetermined Coefficient method. Hint: Solve the homogeneous solution first and then assume ┌ a2 ┐　　┌ a1 ┐ Xp = └ b2 ┘t + └ b1 ┘ 4. (15%) Find the eigenvalues and eigenvectors of this 3-D rotational transformation matrix: ┌ cosθ sinθ 0 ┐ │-sinθ cosθ 0 │ └ 0 0 1 ┘ 5. (15%) Find the eigenvalues and eigenvetors of A and proves A's eigenvectors are orthogonal. ┌ 2 4 -6 ┐ A = │ 4 2 -6 │ └ -6 -6 -15 ┘ 6. (15%) Find the linear system AX = B, where the matrix A is given by ┌ x1 ┐ ┌ -1 5 -1 -3 ┐ │ x2 │ ┌ b1 ┐ A = │ 4 -1 2 6 │, X =│ x3 │, and B = │ b2 │ └ 3 4 1 3 ┘ └ x4 ┘ └ b3 ┘ Find all the possible vectors B for which the linear system has the non-trivial solution. Determine the solution X. --

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