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
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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.
The following facts about a company are known: 1. At present, its total capital is £3 million. 2. It has just sold zero-coupon bonds with the total nominal value of £2 million which it promises to repay in 18 months from now. 3. The total capital F(t) of the company follows the geometric Brownian motion with parameters µ = 0.15 and σ = 0.2. The continuously compounded annual interest rate r = 6%. Within the framework of the Merton model, establish the following. (c)What is the probability that the company would default on its promise to bond holders?
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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