[試題] 104-1 范致豪 工程數學一 第一次小考
課程名稱︰工程數學(ㄧ)
課程性質︰生工系必修
課程教師︰范致豪
開課學院：生農學院
開課系所︰生工系
考試日期（年月日）︰2015/10/14
考試時限（分鐘）：60
試題 :
1. Test for exactness,If exact,solve.If not,use an integrating factor
2
(a) 2xydx+x dy=0 (10%)
(b) sinxcosydx+cosxsinydy=0 (10%)
2. In a room containing 20,000ft^3 of air,600ft^3 of fresh air flows in per
minute,and the mixture (made practically uniform by circulating fans) is
exhausted at a rate of 600 cubic feet per minute(cfm).
(a) What is the amount of fresh air y(t) at any time if y(0)=0 (10%)
(b) After what time will 90% of the air be fresh? (15%)
3. Find the general solution.If an intial condition is given,find also the
particular solution .
(a) y'+ ysinx =e^(cosx) ,y(0)=-2.5 (10%)
(b) xy' +4y =8x^4 ,y(1)=2 (15%)
4. A radioactive material decomposed at a rate proportional to the amount
present.Suppose that the proportional constant is k=-1.5x10^(-11) 1/sec
for the radioactive substance contained in the nuclear waste of our nuclear
power plants .How many years will it take for this substance to decompose to
1/2 of its original quantity? (25%)
(hint:背面附積分公式)
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