Write out and explain your steps for each of the problems. (5) (12 pts) Solve the following partially decoupled first-order system:dx=dtx²y= 2y(6) (12 pts) Find all the equilibrium solutions of the following first-order system:{dxdydt==x² - y²2x+y-3(7) (14 pts) Determine the bifurcation value μo for the following equation with parameter μ. Thendraw the phase lines of the differential equation for μo -1, μ and μo +1.dy = y2 — 2pug-dt(8) (14 pts) Find the solution of the following linear system with the given initial condition:and=3x+2y= 3x + 4y{dx{x(0) = 3y(0) = 2

Problem (5): Solve the partially decoupled system {dtdx=x2ydtdy=2y Step 1: Solve for y(t) The…

Answered: (5) (12 pts) Solve the following…

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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Given that alpha, beta, r and mu > 0. a) Find parameters for alpha, beta, r and mu so that the disease dies out. b) Using a phase line, we see that the model has a positive equilibrium. What is that equilibrium?
Consider the following system of differential equations describing two populations: æ' = 0.02a (1 – ) + 0.004xy 80 (1 y' = 0.04y(1 ) - 0.0002xy 20 a. Describe what the equations tell us about the populations (how the populations behave absent interaction, and how interaction affects each population). b. Find all equilibrium points algebraically. c. Classify each equilibrium using linearization d. Sketch some solution curves in the phase plane (x-y plane)
For the equation dy/dt = y4 + 4y3 − 5y2(a) Find all the equilibrium solutions, and determine their stability type.(b) Sketch the phase line and draw several solution curves in the ty-plane.(c) Determine all the inflection levels, if any at all, for the solutions curves.
Consider the logistic equation dP dt = -KP(1-B) where k> 0 and B> 0 are constants. (a) Find and classify equilibrium solutions and draw a phase line. (b) Use your work from part (a) to sketch the family of solutions corresponding to this differential equation. (c) If P(t) represents a population that is governed by the reverse logistic equation, use your work in part (b) to interpret the behavior of P(t) and explain the possible fates of such a population. (d) Use your results from part (c) to determine the arbitrary constant if P(0) = Po. Is it possible to evaluate the lim P(t)? If so, evaluate the limit, if not, explain why. t-∞0
Solve the related phase plane differential equation for the given system. Then sketch several representative trajectories (with their flow arrows). dx/dt=(8y-8x)(1/9y-1),dy/dt=(8x-8y)(1/9x-1)
Solve the following system and draw a phase portrait
Q.1 Equations (1.1) and (1.2) are linear differential equations for a drug delivery system. + 5x - y = 0 Equation (1.1) Equation (1.2) dx dt dy dt - 4x + 2y = 0 - (a) Is the system homogenous or non-homogenous? State your reason. (b) Derive the two-compartment model represented by the differential equations. (c) Construct a general solution (x(t) and y(t)) to the system. (d) Solve for x(t) and y(t) if the initial conditions at t = 0 (s) are x(0)=1 and y(0)=2, respectively.
2.5 Consider the system x' = (² 9) x. Find the special values of a for which the phase plane changes type. (Assume det A=0).
1) Find all equilibrium solutions of the equation (1 − x) (x² − 4) - x = and classify each one in terms of stability. Draw a phase space diagram and sketch by hand several typical solution curves. Describe the long term (t → ±∞) behavior of the solutions.
Please dont provide handwritten solution .....
For the equation dy/dt = y^4 + 4y^3 - 5y^2 find all equilibrium solutions and determine their stability type sketch the phase line and draw the solution curves in the ty-plane determine all the infelction levels, if any at all for the soltuon curves
Consider the autonomous differential equation dx/dt=-7(x+1)(3-x)(x-5)^2 Find the equilibria, draw a phase line for the differential equation, and then classify the equilibria. Finally, sketch representative solutions to the differential equation based on the phase line.

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