[試題] 102上 陳俊全 常微分方程導論 期中考
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（年月日）︰2013/11/12
考試時限（分鐘）：15:30~17:20
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Choose 5 from the following 6 problems.
(1) Solve the equations.
(a) (1+x)y' + 2y = cosx, y(1) = 0.
4x^2 － 2xy ＋ y^2
(b) y' = -------------------.
2x^2
(2) Solve the equations.
(a) y' + y^2 + y^3 = 0, y(0) = 1.
(b) y' + y/x + x^2 = 1, y(1) = 2.
δf
(3) Let ψ(t) be the solution of y' = f(y), y(0) = 0.5, where —— is
δy
continuous, f(0) = f(1) = 0, f(y) < 0 for 0 < y < 1. Show that
(a) 0 < ψ(t) < 1 for t > 0;
(b) lim ψ(t) = 0.
t→∞
(4) (a) Solve 2y'' + 5y' － 3 = 0, y(0) = 0, y'(0) = 1.
2 '' '
(b) Find all solutions of t y + 4ty + 2 = 0, t > 0.
(5) Consider a pond that initially contains 20 million liters of fresh water.
The water containing an undesirable chemical flows into the pond at the
rate of 10 million(liter/year) and the mixture in the pond flows out at
the same rate. The concentration γ(t) of chemical in the incoming water
varies periodically with time according to the expression γ(t) = 2 +
sin2t (g/liter). Let Q(t) denote the amount of chemical in the pond at
time t. Construct a mathematical model of this flow process and determine
Q(t). Also find the limit of the average amount of Q(t):
1 L
lim — ∫ Q(t)dt.
L→∞ L 0
a
(6) Consider the equation y' = |y(t)| , y(0) = 0.
(a) Prove that the equation has at most one solution if a > 1.
(b) Show that the equation has more than one solution if 0 < a < 1.
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◆ From: 111.185.135.96
※ 編輯: SamBetty 來自: 111.185.135.96 (02/07 19:10)
※ 編輯: SamBetty 來自: 111.185.135.96 (02/07 19:13)