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）︰2012/09/11
考試時限（分鐘）：
試題 :
ODE QUIZ 1 9/11/2012
You have to turn in 1. b), 2. b) 3. d) e) in class.
1.
a) State and prove the fundamental theorem of calculus.
b) Solve the differential equations for t near 0
f'(t) = 1 +2t + 3t^3, f(0) = 1,
f'(t) = sin(st) + 5, f(0) = 7,
f'(t) = 3e^2t + t, f(0) = 4,
1
f'(t) = ln(t + 1) + ────── f(0) = 0.
(2t+1)(t+2)
2. Let M_n(R) be the collection of all n by n matrices. For A ∈ M_n(R), we let
P_A(x) = det(xI_n - A), where I_n is the identity matrix.
n-1
P_A(x) = x^n + Σ a_i˙x^i is called the characteristic polynomial of A. The
i=0
zeros of P_A(x) are called the eigenvalues of A.
a) (Cayley-Hamilton theorem) Show that
n-1
P_A(A) := A^n + Σ a_i˙A^i + a_0˙I_n = (0)_n ×n.
i=1
b) Find eigenvalues for each of the following matrices A ∈ M_n(R) and find a
matrix Q such that (Q^-1)AQ is a diagonal matrix.
╭ 1 4 ╮ ╭ 7 -4 0 ╮
│ │, │ 8 -5 0 │.
╰ 3 2 ╯ ╰ 6 -6 3 ╯
3. For A ∈ M_n(R), we define the matrix norm of A be
|Av|
||A|| := max ──,
V∈R^n\{0} |v|
where |‧| is the Euclidean norm of the vectors in R^n. Now we define
∞
e^A := I_n + Σ A^i.
i=1
a) Show that e^A ∈ M_n(R). (You have to show each element of e^A ∈ R.)
b) Show that if AB = BA, then e^Ae^B = e^(A+B).