youhave been given the followingfollowing autonomousfirst order differential equation.2dy = (y-2) ² (y + 8)dxy=2The critical points areClassify these critical points (in the givenorder) as asymptatically stable,and y =-8custable,or semi-stable.chooseyourgiven below.answerfrom the options@ asymptotically stablestableustable and semi-stable.ustable and asymptotically stable.semi-stable and unstableE asymptatically stabbe and unstablesami-stable and asymptotically stableG semi-stable and semi-stableand asymptoticallyH asymptotically and semi-stableustable and unstable.

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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 differential equation d = 100x(x − 5) – h models a logistic population with harvesting rate h. Determine the dependence of the number of critical points on h and construct a bifurcation diagram.
Use a computer system or graphing calculator to plot a slope field and/or enough solution curves to indicate whether each critical point of the given differential equation is stable, unstable, or semistable (2-4) Identify any semistable critical point(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The differential equation has semistable critical point(s) at x= (Simplify your answer. Use a comma to separate answers as needed.) O B. The differential equation has no semistable citical points.
Match the differential equation with its direction field. y' = 8(x + y) – 1 y Q.2 10 0.1 0.1 -0.3 -0.2-0.1 0.1 0.2 -0,3- I -Q.31 +0,2 -0.1 0.1 0,2 0,3 0.1 //// //// //////// o for o < x < T/8, 0 < y < T/8. O y' = 8(x + y) - 1 = 0 on the line y = -x + 1/8, and y' = -1 on the line y = -x. O y' = 8(x + y) - 1 = 0 on the lines x = 0 and y = 8. O The slopes at each point are independent of y, so the slopes are the same along each line parallel to the y-axis. Note that for y = 8, y' = 0. O The slopes at each point are independent of x, so the slopes are the same along each line parallel to the x-axis. Note that for y = 8, y' = 0.
Consider the autonomous differential equation. y' = (y – 1)(3 – y)² Sketch solutions curves in the t – y plane corresponding to the following initial conditions. Make sure you label your graph and label any equilibrium solutions. y(0) = 4 y(0) = 3 y(0) = 2 y(0) = 1 y(0) = 0. %3D
You can use methods such as variation of parameters, application of exponential shift theorem, and evaluation of some inverse differential operators.
A population P(t) of field mice, where P is the number of mice and t is measured in months from some starting point, grows at the rate of 50% per month. However, owls in the neighbourhood eat them at the rate of 600/month. A model for the mice population is given by the differential equation dP 0.5Р — 600 dt Plot P' vs P, and the phaseline.
Population of the world, (x) is assumed to be a continuous variable at time t years. The rate of increase of x due to birth is ax and the rate of decrease of x due to death is Bx, where a and B are positive constant. a. By applying differential equation concept and assuming a = 0.060 and B = 0.040. Show the population of the world (x) double itself in approximately 35 years. Justify your answer. b. By using appropriate scale, draw the graph of world population growth.
Find the equilibrium points of = e -6y — 9 and classify each one as stable or unstable. dt Stable equilibria occur at y = Unstable equilibria occur at y = (If there is more than one equilibrium of a certain type, enter a comma-separated list. If there are no equilibria, enter "none".) On paper, sketch several solutions to the differential equation.
Which of the following equation/s is/are a separable differential equation? dy dr = y+1 y = xy+1 Option 1 Option 4 e²-2 dy + ydx = 0 (x + y)dx dy = |- Option 3 Option 2
The first-order differential equation y' = y(1 - y) has a stable equilibrium at y=0. has an unstable equilibrium at y-1. has no equilibria. is a linear equation in y. has a stable equilibrium at y-1.
(b) Consider the following phase line. y=2 y=-2 i. Write down a differential equation that has this phase line. ii. Describe the behaviour of the solution that satisfies y(-3) = 0 as the independent variable increases.

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