課程名稱︰常微分方程導論 課程性質︰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（年月日）︰2009/11/10 考試時限（分鐘）：120 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Choose 4 from following 6 problems. 1. Solve the equations. (a) (1 + t^2)y' + 6ty = (1 + t^2)^-1 t, y(1) = 0 (b) y' = (ycosx + y) / (3y + 1) 2. Solve the equations. (Hint: let v = y/x.) (a) y' = (y + 4x) / (x-y) (b) y' = (x^2 - xy + y^2) / x^2 3. Solve the equations. (a) 1 + y'(x/y - e^y) = 0 (Hint: find an integrating factor to make the equation exact.) (b) y' + 5y + y^3 = 0, y(0) = -2 4. Suppose that (δf / δy) (註:δ為偏微分符號) is continuous, f(1) = f(3) = 0 and f(y)(y-1)(y-3) < 0 if y≠ 1,3. Consider the initial value problem y'(t) = f(y(t)) on R, 1 < y(0) < 3. (a) Show that 1 < y(t) < 3 for any t.. (b) Show that lim y(t) = 3 t→∞ 5. Consider the equation y'' - 2y' - 3y = 0 (a) Find all solutions of the equation. (b) Find the solution which satisfies y(0) = 3, y'(0) = 1. (c) Find a solution which satisfies y(0) = 3 and lim y(t) = 0 t→∞ 6. A tank initially contains 100 liters of pure water. A mixture containing a concentration of α g/liter of salt enters the tank at a rate 2 liters/min, and the well-stirred mixture leaves the tank at the same rate. Find an expression in terms of α for the amount of salt in the tank at any time t. Also find the limiting amount of salt in the tank as t → ∞. --

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