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）︰2020/1/8
考試時限（分鐘）：60分鐘
是否需發放獎勵金：是
試題 :
1. Find the recurrence formula for the power series solution around t = 0
for the nonhomogeneous differential equation y''+ ty = e^(t+1).
∞
Given that e^t = Σ (t^n/n!)
n=0
2. True or False?
(a) If Xis an eigenvector of A associated with eignevalue λ and
B = P^(-1)AP holds, then Y = PX is an eigenvector of B associated with
the same eigenvalue.
(b) Eigenvectors corresponding to different eigenvalues are not necessarily
linearly independent.
(c) det A^(T) = det A, provided A is a square matrix; det A^(-1) = det A,
provided A has an inverse.
(d) If A and B are nonsingular, then (AB)^(-1) = B^(-1)A^(-1) and
[A^(T)]^(-1) = [A^(-1)]^(T).
(e) If B is formed from a square matrix A by interchanging two rows or
two columns of A, then det A = -det B.
3. (a) Find a matrix B with eigenvalues λ1 = 1 and λ2 = 0, with their
corresponding eigenvectors x1 = [3,1]^(T) and x2 = [2,1]^(T).
(b) Calculate B^160.
┌ -3 1 ┐ ┌ 3t ┐
4. solve X' = │ │X + │ │ by Variation of Parameters.
└ 2 -4 ┘ └ e^(-t) ┘