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Chemical reactions When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled by the autonomous differential equation
where k > 0 is a constant of proportionality and β > α > 0. Here X(t) denotes the number of grams of the new compound formed in time t.
- (a) Use a phase portrait of the differential equation to predict the behavior of X(t) as t → ∞.
- (b) Consider the case when α = β. Use a phase portrait of the differential equation to predict the behavior of X(t) as t → ∞ when X(0) < α. When X(0) > α.
- (c) Verify that an explicit solution of the DE in the case when k = 1 and α = β is X(t) = α − 1/(t + c). Find a solution that satisfies X(0) = α/2. Then find a solution that satisfies X(0) = 2α. Graph these two solutions. Does the behavior of the solutions as t → ∞ agree with your answers to part (b)?
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Chapter 2 Solutions
Bundle: A First Course in Differential Equations with Modeling Applications, Loose-leaf Version, 11th + WebAssign Printed Access Card for Zill's A ... Applications, 11th Edition, Single-Term
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,