课程名称︰常微分方程导论
课程性质︰必修
课程教师︰夏俊雄
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2012/09/11
考试时限（分钟）：
试题 :
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).