課程名稱︰常微分方程導論 課程性質︰數學系大二必修 課程教師︰陳俊全 開課學院：理學院 開課系所︰數學系 考試日期︰2017年11月17日(五) 考試時限：08:10-10:00，計110分鐘 試題 : Choose 4 from the following 6 problems 1. (25%) Solve the following equations. (a) dy/dx = (y sin(x) - y)/(y^2+1). (b) 2(x^2-y^2)dx + xydy = 0, y(1) = 2. (c) y sin(2x)dx + (sin(x))^2 dy = 0. 2. (25%) Solve the equations. (a) y'' + 2y' + 7y = 0, y(0) = 1, y'(0) = -2. (b) y'' - 2y' + y = e^t. (c) y'' - 2y' + y = e^t/(t^2+1). (Hint: variation of parameters.) 3. (25%) (a) Prove that the initial value problem y'(t) = y /(1+t^2 + y^2), y(0) = 1. has at most one solution. (Hint: Gronwall's inequality.) (b) Show that the initial value problem y' = y^(1/5), y(0) = 0 has more than one solution. 4. (25%) Suppose that the population y of a certain species of fish is described by the Schaefer model with harvesting dy/dt = r(1 - y/K)y - Ey, y(0) = Y_0, where r, K and E are positive constants and E < r. (a) Show that there are two equilibrium points y1 = 0 and y2 = K(1 - E/r). (b) Show that if 0 < Y_0 < y2, then y(t) is increasing for t>0 and lim y(t) = y2. t->∞ (c) Show that y = y2 is asymptotical stable and y = y1 is ubstable. 5. (25%) Let t ψ (t) = 0 , ψ (t) = ∫sψ (s)ds + 1, n = 0,1,2,3,... 0 n+1 0 n (a) Show that 0≦ψ (t)≦2 for 0≦t≦1. n (b) Show that lim ψ (t) exists for 0≦t≦1. n→∞ n (c) Find lim ψ (t) for 0≦t≦1. n→∞ n 6. (25%) A swimming pool whose volume is 10,000 gal contains water that is 0.02% chlorine. Starting at t = 0, city water containing 0.001% chlorine is pumped into the pool at a rate of 5 gal/min. The pool water flows out at the same rate. What is the percentage of chlorine in the pool after 1 hour? When will the pool water be 0.004% chlorine? --

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