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2.12 In autocatalytic reactions, the rate of reaction is increased by the presence of the substance being produced. Suppose that in an autocatalytic reaction a chemical X is being converted into a chemical Y with x and y as the concentrations of X and Y, respectively. Assume that the rate of change of X is proportional to the product xy. If x (0) = a, y(0) = 3 and a x(t) = y(t) — ß are known, show that x satisfies the initial value problem x' -XX = -kx(3+a-x), x(0) www = where k is a positive constant. Find its solution and show that lim, ox(t) = 0, lim y(t) = a + ß.

2.12 In autocatalytic reactions, the rate of reaction is increased by the presence of the substance being produced. Suppose that in an autocatalytic reaction a chemical X is being converted into a chemical Y with x and y as the concentrations of X and Y, respectively. Assume that the rate of change of X is proportional to the product xy. If x (0) = a, y(0) = 3 and a x(t) = y(t) — ß are known, show that x satisfies the initial value problem x' -XX = -kx(3+a-x), x(0) www = where k is a positive constant. Find its solution and show that lim, ox(t) = 0, lim y(t) = a + ß.

Advanced Engineering Mathematics
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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2.12 In autocatalytic reactions, the rate of reaction is increased by the presence of the substance being produced. Suppose that in an autocatalytic reaction a chemical X is being converted into a chemical Y with x and y as the concentrations of X and Y, respectively. Assume that the rate of change of X is proportional to the product xy. If x(0) = a, y(0) = 3 and a-x(t) = y(t)-B are known, show that x satisfies the initial value problem x' = kx(3+a-x), x(0) = a where k is a positive constant. Find its solution and show that lim, ox(t) = 0, lim o y(t) = a + ß.
Transcribed Image Text:2.12 In autocatalytic reactions, the rate of reaction is increased by the presence of the substance being produced. Suppose that in an autocatalytic reaction a chemical X is being converted into a chemical Y with x and y as the concentrations of X and Y, respectively. Assume that the rate of change of X is proportional to the product xy. If x(0) = a, y(0) = 3 and a-x(t) = y(t)-B are known, show that x satisfies the initial value problem x' = kx(3+a-x), x(0) = a where k is a positive constant. Find its solution and show that lim, ox(t) = 0, lim o y(t) = a + ß.
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