課程名稱︰常微分方程導論
課程性質︰系必修
課程教師︰林紹雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2010年10月26日
考試時限（分鐘）：50分鐘
是否需發放獎勵金：是
試題 :
There are problems (a) to (b) with a total of 50 points. Please write down yo-
ur computational or proof steps clearly on the answer sheets.
A real 4 ×4 matrix A has characteristic polynimial (x^2 +2x +2)^2. Its eigen-
value -1+i (where i = (-1)^(1/2)) has generalized eigenvectors given by
┌ 1+i ┐ ┌ 1+i ┐
w1 = ｜ i ｜ , w2 = ｜ 1+i ｜
｜ 0 ｜ ｜ 1+i ｜
└ 0 ┘ └ i ┘
so that Aw1 =(-1 + i)w1 and Aw2 = (-1+i)w2 + w1.
(a) (35 points) Write down a real basis for the solution space of the homogen-
eous equation y' = Ay.
(b) (15 points) Consider the inhomogeneous equation y' = Ay + we^(-t)(cos t),
where w ∈ R^4. What form of a particular solution is given by the method
of undetermined coeffcients?
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