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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