Differential equations : Solution set
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
This is a series of problems involving differential equations including word problems and particular solutions.
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