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?) --

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