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ch02_3 - Ch 2.3 Modeling with First Order Equations...

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Ch 2.3: Modeling with First Order Equations Mathematical models characterize physical systems, often using differential equations. Model Construction : Translating physical situation into mathematical terms. Clearly state physical principles believed to govern process. Differential equation is a mathematical model of process, typically an approximation. Analysis of Model : Solving equations or obtaining qualitative understanding of solution. May simplify model, as long as physical essentials are preserved. Comparison with Experiment or Observation : Verifies solution or suggests refinement of model.
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Example 1: Mice and Owls Suppose a mouse population reproduces at a rate proportional to current population, with a rate constant of 0.5 mice/month (assuming no owls present). Further, assume that when an owl population is present, they eat 15 mice per day on average. The differential equation describing mouse population in the presence of owls, assuming 30 days in a month, is Using methods of calculus, we solved this equation in Chapter 1.2, obtaining 450 5 . 0 - = p p t ke p 5 . 0 900 + =
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Example 2: Salt Solution (1 of 7) At time t = 0, a tank contains Q 0 lb of salt dissolved in 100 gal of water. Assume that water containing ¼ lb of salt/gal is entering tank at rate of r gal/min, and leaves at same rate. (a) Set up IVP that describes this salt solution flow process. (b) Find amount of salt Q ( t ) in tank at any given time t . (c) Find limiting amount Q L of salt Q ( t ) in tank after a very long time. (d) If r = 3 & Q 0 = 2 Q L , find time T after which salt is within 2% of Q L . (e) Find flow rate r required if T is not to exceed 45 min.
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Example 2: (a) Initial Value Problem (2 of 7) At time t = 0, a tank contains Q 0 lb of salt dissolved in 100 gal of water. Assume water containing ¼ lb of salt/gal enters tank at rate of r gal/min, and leaves at same rate.
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