課程名稱︰常微分方程導論
課程性質︰必修
課程教師︰夏俊雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2012/09/25
考試時限（分鐘）：
試題 :
ODE QUIZ 9/25/2012
You have to turn in the first problem set in class.
1. Solve the following differential equations.
x''(t) - x'(t) - 2x(t) = e^t + cost, x'(0) = 1, x(0) = 0.
x'''(t) - 3x''(t) + 3x'(t) - x(t) = 2t + e^t, x''(0) = 2, x'(0) = 2, x(0) = 1.
x''(t) - 4x'(t) + 4x(t) = te^2t + 1, x'(0) = 0, x(0) = 1.
x''(t) + tx'(t) + 2x(t) = 0, x'(0) = 2, x(0) = 1.
x''(t) + x(t) = sin(2t) + cost, x(0) = 1, x'(0) = 0.
2.For a matrix A ∈ M_n(R), the minimal polynomial of A is a nonzero polynomial
p(x) such that
a) p(A) = 0, and
b) for any nonzero polynomial f(x) satisfying f(A) = 0, deg(p(x))≦deg(f(x)).
Show the following statements.
(1) If p(x) is a minimal polynomial of A and the polynomial f(x) satisfying
f(A) = 0, then p(x)|f(x) (p(x) is a factor of f(x)).
(2) Let
╭ 0 1 0 0 … 0 ╮
│ 0 0 1 0 … 0 │
│ 0 … … … … 0 │
A = │ 0 … … … … 0 │
│ 0 … … … 0 1 │
╰-a_0 -a_1 -a_2 -a_3 … -a_n-1╯
Prove that p(x) = x^n + a_n-1˙x^n-1 + … + a_1˙x + a_0 is a minimal
polynomial of A.