Differential equations
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1. A tank with a capacity of 500 gal. originally contains 200 gal. of water with 100 lb. of salt in solution. Water containing 1 lb. of salt per gallon is entering at the rate of 3 gal./min. and the mixture flows out at a rate of 2 gal./min.
(i) Write a differential equation for the concentration in the tank before the tank overflows.
(ii) How many minutes before the tank starts to overflow?
2. Solve the initial value problem
4y"-y = 0, y(0) = 2, y'(0)=Q
and find Q so that the solution approaches zero as t approaches infinity
3 . Solve y"?6y'+9y = 0 with y(0) = 0, y'(0) = 2 ..
4 y"-y'?2y = 0, y(0) = a, y'(0) = 2 . Find a so that the solution approaches zero
as t approaches infinity
.
5. If y, (x) =ex is a solution of (x ? l) y"--xy' +y = 0 x > 1, find another linearly independent solution.
6. A pond containing 1,000,000 gal. of water is initially free of a certain undesirable chemical. Water containing 0.01 gms/gal. of the chemical flows into the pond at a rate of 300 gal./hr., and water also flows out of the pond at the same rate. Assume that the chemical is uniformly distributed throughout the pond.
(a) Let Q(t) be the amount of the chemical in the pond at time t . Write down an initial value problem for Q(t) .
(b) At the end of 1 year the source of the chemical in the pond is removed; thereafter pure water flows into the pond, and the mixture flows out at the same rate as before. Write down the initial value problem that describes this new situation.
(c) "Sketch" Q(t) .
7. Find the particular solution to the equation 2y"+3 y'+y = t 2 + 3 sin t
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Solution Summary
This is a series of problems involving differential equations including word problems and particular solutions.
Education
- BSc , Wuhan Univ. China
- MA, Shandong Univ.
Recent Feedback
- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"
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