各位数学高手请进，DE问题。。。！！！

zxteh

小弟有一些问题向要想各位数学高手请进....


问题一：
A 1500 gallon tank initially contains 600 gallons of water with 5 lbs of salt dissolved in it. Water
enters the tank at a rate of 9 gal/hr and the water entering the tank has a salt concentration of
1/5（1+(cos(t)) lbs/gal.
If a well mixed solution leaves the tank at a rate of 6 gal/hr, how much salt is in the tank when it
overflows?

[Hint : set up the initial value problem, find Q(t), for the flow rate of the water entering (given), the
concentration of the salt in the water entering (we’ve got that), the flow rate of the water leaving
(given) and the concentration of the salt in the water exiting (not given).]

谢谢各位的帮忙，小弟感激不尽。。。！！！

hihi23

Let S(t) be the amount of salt at time t

S'(t)=(flow of salt into tank at time t) - (flow of salt out of tank at time t)
       =9 x 1/5（1+(cos(t))  - 6x (concentration of solution)
       = 9/5（1+(cos(t)) - 6x(amount of salt in tank at time t / volume of the solution at time t)
       =9/5（1+(cos(t)) -6 x (S(t) / (600+3t))
(volume =600+3t becoz initial volume =600  and increase 3 for every hour)

S'(t) =9/5（1+(cos(t))  - [2/(200+t)] x S(t)   ----- (1)
S(0)=5                                                           ----- (2)

(1)==>  S'(t)+[2/(200+t)] x S(t) =9/5 [1+(cos(t))]            (linear DE)

integrate [2/(200+t)] = ln(200+t)^2
integrating factor is (200+t)^2

d/dt [(200+t)^2 x S(t)]= 9/5 [1+(cos(t)] [(200+t)^2]


[(200+t)^2 x S(t)]= integrate {9/5 [1+(cos(t)] [(200+t)^2]} dt




之后就把积分里的那堆东西展开，再一个一个积分，再把S(0)=5 代进去，就找到S(t) 的 equation 了 。中间的过程应该要用到 integration by part.... 复杂。

至于 t 要代多少应该可以算得到。

zxteh

原帖由 hihi23 于 16-2-2009 08:42 PM 发表
Let S(t) be the amount of salt at time t

S'(t)=(flow of salt into tank at time t) - (flow of salt out of tank at time t)
       =9 x 1/5（1+(cos(t))  - 6x (concentration of solution)
       = 9/5 ...

问题二：

zxteh

原帖由 zxteh 于 25-2-2009 02:01 PM 发表


问题二：

没人帮小弟吗？

中庸

第一題好像是那閒大學的tutorial來的嗎？