1. Consider the differential equationdx(t) = -x³.First, use a geometric argument to show that x = 0 is a locally stable equilibrium. Next,perform linear stability analysis of x = 0. What, if anything, can you conclude regardingthe rate of convergence of x(t) towards x = 0?Finally, solve the differential equation explicitly and show thatlim x(t) = 0for any initial condition. Does the analytical solution of the ODE agree with your linearstability analysis?

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Answered: 1. Consider the differential equation d…

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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The population P = P(t) of a species satisfifies the logistic equation P'= aP(1-bP), P(0) = P0 > 0. (a) Find P for t > 0. (b) Find the limit of P(t) as t → ∞. What does this mean in terms of the population?
(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.
[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
Id=20/36
Consider the differential equation y = y(3-y) (5-y)". Identify all equilibrium solutions. Sketch the direction field for the differential equation. If y(t) denotes the solution of the differential equation satisfying the initial condition y(0) = 4, find the limit lim y(t).
Find the steady periodic solution X sp(t) = C cos (@t-α) of the given equation mx'' + cx' + kx = F(t) with periodic forcing function F(t) of frequency @. Then graph xsp (t) together with (for F(t) comparison) the adjusted function F₁ (t)= mw x'' + 12x' + 11x = 9 cos 2t
(4) This exercise is intended to be an initial exposure for taking derivatives with a different variable other than r. The same rules apply, just use the given variable instead of r. (y? + 8)* (a) If f(y) = find f'(y). (b) If z = 9t?e-4", find (c) If k(8) = In (64 + 0), find [k(8)].
(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 1. (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. (f) Plot the potential function for both signs of the free parameter, positive (B= 0.5 and = 1) and negative (B=-0.5 and B=-1). Find extrema of the potential function and their positions on the -axis. In which cases are the extrema stable and unstable?
Let f(x, y) = cos(x² − y), and let x = g(t) and y = h(t) be differentiable functions such that

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