[試題] 99上 林紹雄 常微分方程導論 第一次小考
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
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.251.220