Sketch phase lines and graph the solutions that satisfy the given initial condition on a single pair of axes. The given differential equation is:\[\frac{dy}{dx} = -x^2 - 2x - 2\]with initial conditions:- \( y(0) = 0 \)- \( y(1) = 1 \)- \( y'(0) = 2 \) This equation represents a first-order ordinary differential equation with specific initial values provided to determine a unique solution. There is no graph or diagram included in this image.

Given: dydx=-x2-2x-2, y(0)=0, y(1)=1, y(0)=2

Answered: - Zm – 2, y{0; = 1, 3(1)=1,/0;= 2

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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(a) Write down the equation of motion for the point particle of mass m moving in the Kepler potential U(x) = -A/x+ B/x² where x is the particle displacement in m. (d) Mimimise the number of independent parameters in the modified equation of motion with dissipation by introducing dimensionless varibles and r for x and r. (e) Present the equation derived in the form of an energy balance equation and find the expressions for the effective kinetic and potential energies (7 and V), and dissipative function F.
2. Assume that the population of a colony of Brazilian fire ants, P, is described by the function P(t)= \t+1·e-0.5t is measured in days (t=0 when one begins monitoring the population). (a) Find the initial population, P(0) (include units with your answer). +10t +200. The population here is measured in thousands of individuals, and time, t, %3D %3D (b) Find P (t). (c) Evaluate P (0) (include units with your answer).
Justify and analyze the model proposed in equation (2), covering in particular the following issues: an analysis of the validity of equation (2) as a model for the fish population in a fish farm; an analysis of whether, based on model equation (2), we may expect a stable fish population from which harvesting could take place; and, if so, an analysis of how large an initial population of fish will be required to obtain this stable population and the length of time required for the stable population to be established (numerical solver recommended).

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