課程名稱︰常微分方程導論 課程性質︰系必帶 課程教師︰林紹雄 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2010年09月28日 考試時限（分鐘）：50分鐘 是否需發放獎勵金：是 試題 : There are problems A to C with a total of 50 points. Please wirte down your computational or proof steps clearly on the answer sheets. A. Solve the following first0order ODE. Each has 10 points. dy y - x^2 (a) ---- = --------- , y(0) = 0 . Is the solution unique? dx x^2 - y (b) (3x^2 - y^2)dx + (xy - x^3 * (y^(-1)))dy = 0. B. (15 points) Suppose a brine containing 0.2 kg of salt per litre runs into a tank initially filled with 500L of water containing 5 kg of salt. The brine enters the tank at a rate of 5L/min.The mixture, kept uniform by stirring, is flowing out at a rate of 5 L/min. However, after 10 min., a leak develo- pes in the tank and an additional litre per minute of mixture flows out of the tank (see the Figure below). What will be the concentration, in kilogr- am per litre, of salt in the tank 20 min after the leak developse? C. Let y1(t) and y2(t) be two solutions of y' = f(y). Assume that both y1(t) and y2(t) are defined for all t ∈R. Determine which of the following statements is true. Prove your answer. Each has 5 points. (a) If f(y) = y^2 , then y1(t) = y2(t) for all t. (b) If f(y) = 2√y , and y1(1) = y2(t) = 1, then y1(t) = y2(t) for all t. (c) If f(y) = 3(y)^(1/3) , and y1(1) = y2(1) = 1, then y1(t) = y2(t) for all t ----------------| 5 L/min →　　　　　　 --- 0.2 kg/L → 　　　　　　 　　→ 5 L/min ---|　　　　　　--- 　　　　　　|______ 　__| 　　　　　　　　　|↓↓| 　　　　　　　　 1 L/min --

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