Derive the solution of the logistic initial value problemP'(t) = kP(t) (M - P(t)), P(0) = Po,in terms of t, M, and Po. Describe the asymptotic behaviour of P(t), and sketch the phase diagramassociated with the equilibrium solutions of the above initial value problem.Problem 4

The logistic equation can be solved using the separation of variables technique.

Answered: Derive the solution of the logistic…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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[11] For each of the following equations, find general solutions; solve the initial value problem with initial condition x₁ (0) = -1, x₂ (0) = 2; • sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. -2 -13 dx₁/dt = x₁ + x2 x₁ = x₁4x₂ (a) x' = [ X (b) { 4 dx2/dt x₂ = 5x₁²x₂ 1 = -2x1 - x2
(4) (a) Derive the solution MPO Po+(M-Po)ek Mt P(t): of the extinction-explosion initial value problem P' = kP(P-M), P(0) = Po- (b) How does the behavior of P(t) as t increases depend on whether 0 M?
Solve the initial value problems in Problems and graph each solution function x(1). x" + 4x' + 4x = 1+ 8(t – 2); x(0) = x'(0) = 0
[13] For each of the following equations, find general solutions; • solve the initial value problem with initial condition x₁(0) = -1, x₂ (0) = 2; • sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. x₁ +4x₂ -5 0 (a) =[ 1 d x1 dt X2 () = »E]-[73][B] ©*-[72]* (c) = 4 -x1 + 5x₂ - X2 0
2. The Solow Model Suppose the production function is given by Y, = 0.5/K, VN Y, (1) Transform the production function in per worker terms, i.e. write down as a N K, function of N (2) Denote the saving rate by s, the depreciation rate by 8, solve for the K* steady-state capital per worker the steady-state level of output per worker and the steady-state consumption per worker as functions of s and 8. (3) Suppose 8 = 0.05, compute the steady-state output per worker and the steady-state consumption per worker for s= 0; s= 0.1; s=0.2; and s =1. Explain the intuition behind your results.
Problem #8: Solve the following initial value problem. y"-20y + 149y = 0, y(0) Problem #8: = 4, y'(0) = 10 Enter your answer as a symbolic function of x, as in these examples Do not include 'y=' in your answer.
1. In a company a worker must make a total of 320 pieces per day as a minimum production target. Francis makes 180 pieces on his first day at work, after 3 days he makes 250 pieces , if the number of parts produced increases at a proportional rate to those missing to produce to complete the minimum target required, a) Propose a Differential Equation for the number of parts produced with respect to time and use it to determine if at 10 days it already meets the minimum target required
The number N(t) of supermarkets throughout the country that are using a computerized checkout system is described by the initial-value problem dN = N(1 dt 0.0004N), N(0) = 1. - (a) Use the phase portrait concept of Section 2.1 to predict how many supermarkets are expected to adopt the new procedure over a long period of time. supermarkets By hand, sketch a solution curve of the given initial-value problem. N 2500 2500 1200 1000 2000 2000 800 1500 1500 600 1000 1000 400 500 500 200 10 15 20 10 15 20 10 15 20 N 1200 1000 800 600 400 200 t 20 10 15 (b) Solve the initial-value problem and then use a graphing utility to verify the solution curve in part (a). N(t) = How many supermarkets are expected to adopt the new technology when t = 10? (Round your answer to the nearest integer.) supermarkets
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Derive the solution of the logistic initial value problem P' (t) = kP(t) (M - P(t)), P(0) = Po, in terms of t, M, and Po. Describe the asymptotic behaviour of P(t), and sketch the phase diagram associated with the equilibrium solutions of the above initial value problem. Problem 4