**Population with Carrying Capacity Problem**Consider a population scenario with the following parameters:- **Birth Rate:** 30/1000 per year- **Death Rate:** 25/1000 per year- **Initial Population:** 100,000Introducing a **Carrying Capacity:** 1,000,000 people### (a) Derive the Logistic Differential EquationTo model the population with carrying capacity, use the logistic growth equation which typically takes the form:\[ \frac{dP}{dt} = rP \left(1 - \frac{P}{K}\right) \]Where:- $ P $ is the population size- $ r $ is the intrinsic rate of increase (birth rate minus death rate)- $ K $ is the carrying capacityGiven:- Birth rate = 30/1000 = 0.03- Death rate = 25/1000 = 0.025Intrinsic rate of increase: $ r = 0.03 - 0.025 = 0.005 $Carrying capacity: $ K = 1,000,000 $Hence, the logistic differential equation is:\[ \frac{dP}{dt} = 0.005P \left(1 - \frac{P}{1,000,000}\right) \]### (b) Derive the first 4 terms of the Taylor Series centered about $ t = 0 $Without solving the differential equation directly, derive its derivatives to formulate the first four terms of the Taylor series.The Taylor series expansion for $ P(t) $ at $ t = 0 $ is given by:\[ P_3(x) = \sum_{k=0}^{3} \frac{f^{(k)}(0)}{k!}(x - 0)^k \]Where $ f^{(k)}(0) $ represents the k-th derivative of $ f(x) $ evaluated at $ x = 0 $.### (c) Use the process of separation of variables to find the solutionTo solve the differential equation using separation of variables:1. Start with the form of the logistic differential equation:\[ \frac{dP}{dt} = rP \left(1 - \frac{P}{K}\right) \]2. Separate variables $ P $ and $ t $:\[

This question is about application of differential equations

3. Consider the Population with carrying capacity problem: Birth Rate = 30/1000 per year and death rate = 25/1000 per year, initial population of 100000 Introduce Carrying Capacity 1000000 people Derive the Logistic differential equation b. Without solving the differential equation, take derivatives of the differential equation and come up with the first 4 terms of the Taylor series centered about t=0. %3D 3 (k) P3 (x) = k! k=0 Use the process of separation of variables to find the solution.

3. Consider the Population with carrying capacity problem: Birth Rate = 30/1000 per year and death rate = 25/1000 per year, initial population of 100000 Introduce Carrying Capacity 1000000 people Derive the Logistic differential equation b. Without solving the differential equation, take derivatives of the differential equation and come up with the first 4 terms of the Taylor series centered about t=0. %3D 3 (k) P3 (x) = k! k=0 Use the process of separation of variables to find the solution.

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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