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）︰2012/11/02
考試時限（分鐘）：100分鐘
試題 :
ODE Midterm 11/2/2012
1. (20 points) Solve the following differential equation.
(4)
y + 5y'' + 4y = 1 - u_π(t), y'''(0) = y''(0) = y'(0) = y(0) = 0,
where u_π(t) is a heavy side function.
2. (10 points) Find the inverse Laplace transform for the following function.
s^3 - 2s^2 - 6s - 6
g(s) = ──────────.
(s^2 + 2s + 2)s^2
3. (20 points) Solve the following problems by power series.
y'' - xy' -y = 0, y(0) = 2, y'(0) = 1.
(2 + x^2)y'' - xy' + 4y = 0, y(0) = -1, y'(0) = 3.
4. (20 points) Solve the following problems by power series.
x''(t) - 4x'(t) + 4x(t) = (t^2)e^(2t) + 1, x'(0) = 0, x(0) = 1.
x''(t) + tx'(t) + 2x(t) = 0, x'(0) = 2, x(0) = 1.
5. (20 points) True or False. Prov or disprove the following statements,
Consider the differential equation
╭
∣ x'(t) = f(x,t)
╱ (0.1)
╲
∣ x(0) = 0,
╰
where f(x,t) is a continuous function define on [-1,1] ×[-1,1]
with |f(x,t)| ≦ 1 for all (x,t) ∈ [-1,1] ×[-1,1].
(a) The maximum interval of existence of this problem is [-0.5,0.5].
(b) There always exists a unique solution to this equation.
6. (20 points) Show that the following differential equation has a unique time
periodic solution.
x'(t) = 2x + sin(3t) + cos(t/2).
(Hint: What is the period of this time periodic solution?)