Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Please solve 1 and 2

**Problem 1: Differential Equation Solution** 1. Solve \( y' = x(y - 3) \). Steps: - Start with \(\frac{dy}{dx} = x(y - 3)\). - Separate the variables: \(dy = x(y - 3)dx\). - Integrate both sides: \(\int \frac{dy}{y - 3} = \int x \, dx\). - The left side becomes \(\ln|y - 3| = \frac{x^2}{2} + C\), where \(C\) is the constant of integration. - Exponentiate to solve for \(y\): \(y - 3 = e^{\frac{x^2}{2} + C} = Ce^{\frac{x^2}{2}}\). - Finally, express \(y\) as \(y = Ce^{\frac{x^2}{2}} + 3\). **Problem 2: Exponential Growth/Decay Model** 2. Pesticide was given to a colony of 8000 ants. If there were 2000 ants left after 2 weeks, find the number of ants after 3 weeks. - The population model is \( y = Ce^{kt} \). **Explanation:** In Problem 1, we solved a differential equation using separation of variables and integration. The resulting function describes a relationship between \(y\) and \(x\). In Problem 2, we are given an exponential decay model where a population of ants is declining over time \(t\). The constants \(C\) and \(k\) need to be determined based on initial conditions and data, typically involving solving for particular values at known time points.
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Transcribed Image Text:**Problem 1: Differential Equation Solution** 1. Solve \( y' = x(y - 3) \). Steps: - Start with \(\frac{dy}{dx} = x(y - 3)\). - Separate the variables: \(dy = x(y - 3)dx\). - Integrate both sides: \(\int \frac{dy}{y - 3} = \int x \, dx\). - The left side becomes \(\ln|y - 3| = \frac{x^2}{2} + C\), where \(C\) is the constant of integration. - Exponentiate to solve for \(y\): \(y - 3 = e^{\frac{x^2}{2} + C} = Ce^{\frac{x^2}{2}}\). - Finally, express \(y\) as \(y = Ce^{\frac{x^2}{2}} + 3\). **Problem 2: Exponential Growth/Decay Model** 2. Pesticide was given to a colony of 8000 ants. If there were 2000 ants left after 2 weeks, find the number of ants after 3 weeks. - The population model is \( y = Ce^{kt} \). **Explanation:** In Problem 1, we solved a differential equation using separation of variables and integration. The resulting function describes a relationship between \(y\) and \(x\). In Problem 2, we are given an exponential decay model where a population of ants is declining over time \(t\). The constants \(C\) and \(k\) need to be determined based on initial conditions and data, typically involving solving for particular values at known time points.
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