课程名称︰工程数学(ㄧ)
课程性质︰生工系必修
课程教师︰范致豪
开课学院：生农学院
开课系所︰生工系
考试日期（年月日）︰2015/10/14
考试时限（分钟）：60
试题 :
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:背面附积分公式)