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15с) — Зс2 — сз %3D 3 800 -Зс, + 18с, — бсз 3D 1 200 —4cт — С2 + 12с, 3D 2 350 |

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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You have measured the water level of a reservoir at different times throughout an year. You need to find the minimum water level that the reservoir had in that year based on these measurements. To work this out which numerical methods would you need and which order would you need to apply them in? O a. Solving a system of linear equations followed by solving an ODE. b. Curve fitting, followed by solving an ODE. c. Solving an ODE followed by root-finding applied to the derivative of the resultant curve. Od. Root finding followed by numerical integration. e. Numerical integration followed by solving a system of linear equations. O f. Curve fitting, followed by numerical integration. O g. Curve fitting followed by root-finding applied to the derivative of the fitted curve. Oh. Numerical integration followed by solving an ODE. UND DND OXO
1. Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains 150 liter (L) of a dye solution with a concentration of 3 g/L. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of 3 L/min, the well- stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches 2% of its original value.
6. A 50 gallons tank initially contains 10 gal of fresh water. At t = 0, a brine solution containing 1 lb of salt per gallon is poured into the tank at the rate of 4 gal/min, while the well-stirred mixture leaves the tank at the rate of 2 gal/min. Find a. the amount of time required for overflow to occur b. the amount of salt in the tank at the moment of overflow
solve and draw a diagram of the problem Two large tanks, each holding 100L of liquid, are interconnected by pipes, with the liquid flowing from tank A into tank B at a rate of 3L/min and from tank B into tank A at a rate of 1L/min. The liquid inside each tank is kept well stirred. Pure water flows into tank A at a rate of 6L/min. The solution flows out of the system from tank A at 4L/min and from tank B at 2L/min. If, initially, tank A contains pure water and tank B contains 20kg of salt, determine the mass of salt in each tank at time t ≥ 0.
H9.
9. Imagine a medieval world. In this world a Queen wants to poison a King, who has a wine keg with 500 L of his favorite wine. The Queen gives a conspirator a liquid containing 5 g/L of poison, which must be poured slowly into the keg at a rate of 0.5 L/min. The poisoner must also remove the well-stirred mixture at the same rate, so that the keg is not suspiciously full. (a) Find a formula for the amount of poison in the keg at any time, measured from the start of the pouring by the poisoner. (b) A plot is hatched for the King to drink wine from the keg while he is on a hunt, where he will become so addled that his prey will surely kill him. The poisoner must pour for a time T, when the poison in the keg reaches a dangerous concentration of 0.005 g/L. Find T. (c) The Lord High Inquisitor of the Realm never learned about differential equations. Nonetheless, knowing the basic numbers (keg size, poison concentrations, etc.), he can produce an estimate for the time T that the poisoner was…
A large vat contains 100 liters of pure water. Salt and water are then added to the tank: salt at a rate of 8 ounces per minute, and water at a rate of 5 liters per minute. Simultaneously, the saltwater solution is drained from the bottom of the vat at a rate of 5 liters per minute. Assume the saltwater is kept thoroughly mixed.a. Write an initial-value problem that gives the amount of salt in the vat after t minutes.(hint: your answer should be separable so that you can solve in part (b)) b. Solve the initial-value problem you gave in part (a).
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