For each of the systems in Problems
(a) Find all the critical points (equilibrium solution).
(b) Use a computer to draw a direction field and phase portrait for the system.
(c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type.
(d) Describe the basin of attraction for each asymptotically stable critical point.
Want to see the full answer?
Check out a sample textbook solutionChapter 7 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Additional Math Textbook Solutions
The Heart of Mathematics: An Invitation to Effective Thinking
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
A Problem Solving Approach to Mathematics for Elementary School Teachers (12th Edition)
Finite Mathematics & Its Applications (12th Edition)
Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)
- (b) A dynamical system is governed by two equations: (a) Find critical points of this system. |x=y, [y=In(x² + y)-3y. Here a dot on the top of a symbol stands for the derivative with respect to t. (c) Using linearisation of the system in the neighbourhood of each critical point, determine the nature of the critical points. Draw qualitatively but neatly these critical points and corresponding trajectory diagrams.arrow_forwardDraw the phase diagram of the system; list all the equilibrium points; determine the stability of the equilibrium points; and describe the outcome of the system from various initial points. You should consider all four quadrants of the xy-plane. All the following must be included, correct and clearly annotated in your phase diagram: The coordinate axes; all the isoclines; all the equilibrium points; the allowed directions of motion (both vertical and horizontal) in all the regions into which the isoclines divide the xy plane; direction of motion along isoclines, where applicable; examples of allowed trajectories in all regions and examples of trajectories crossing from a region to another, whenever such a crossing is possible.arrow_forwardDraw the phase diagram of the system; list all the equilibrium points; determine the stability of the equilibrium points; and describe the outcome of the system from various initial points. You should consider all four quadrants of the xy-plane. All the following must be included, correct and clearly annotated in your phase diagram: The coordinate axes; all the isoclines; all the equilibrium points; the allowed directions of motion (both vertical and horizontal) in all the regions into which the isoclines divide the xy plane; direction of motion along isoclines, where applicable; examples of allowed trajectories in all regions and examples of trajectories crossing from a region to another, whenever such a crossing is possible.arrow_forward
- 2.An engineer at a semiconductor company wants to model the relationship between the device HFE (y) and three parameters: Emitter-RS (x1), Base-RS (x2), and Emitter-to- Base RS (x3). The data are shown in the following table. X1 X2 X3 X1 X2 X3 y 14.620 226.000 7.000 128.400 15.500 230.200 5.750 97.520 15.630 220.000 3.375 52.620 16.120 226.500 3.750 59.060 14.620 217.400 6.375 113.900 15.130 226.600 6.125 111.800 15.000 220.000 6.000 98.010 15.630 225.600 5.375 89.090 14.500 226.500 7.625 139.900 15.380 229.700 5.875 101.000 15.250 224.100 6.000 102.600 14.380 234.000 8.875 171.900 16.120 220.500 3.375 48.140 15.500 230.000 4.000 66.800 15.130 223.500 6.125 109.600 14.250 224.300 8.000 157.100 15.500 217.600 5.000 82.680 14.500 240.500 10.870 208.400 15.130 228.500 6.625 112.600 14.620 223.700 7.375 133.400 (a) Fit a multiple linear regression model to the data. (b) Predict HFE (y) when x1 =14.5, x2 = 220, and x3 = 5.0.arrow_forward(3) The approximate enrollment, in millions between the years 2009 and 2018 is provided by a linear model Y3D0.2309x+18.35 Where x-0 corresponds to 2009, x=1 to 2010, and so on, and y is in millions of students. Use the model determine projected enrollment for the year 2014. 近arrow_forwardNeed help with this Linear First Order Mixing Problem. Thank you!arrow_forward
- Question 2. Find the equilibrium solutions of the SIR Model.arrow_forward2. Suppose that W is the population size of Yellowstone wolves and E is the population size elk in Yellowstone. The equations and dE dt = 0.3E-0.4WE dW dt model the interaction between these species. -0.2W +0.1WE If the elk population is zero, what does the second equation tell you? If the wolf population is zero, what does the first equation tell you? • Divide one equation by the other, relating dW and dE.arrow_forwardA mass weighing 16 pounds is attached to a spring, stretching it 2 feet. A damping mechanism provides a resistance numerically equal to b times the instantaneous velocity. The mass is pulled down 1 foot below equilibrium and released from rest. a. Find the equations of motion for b = 3, b = 4, b = 5. b. Determine if each system is underdamped, critically damped, or overdamped.arrow_forward
- plot phase portrait of the following Nonlinear system. x₁ = 29₂2₂ - XG X₂₂ 1x²2₂² = -296²³-96₂arrow_forward2a. Find a change of variable that transforms the equation into an autonomous equation change of variable: new equation: b. Sketch the phase line for the resulting equation and use it to sketch graphs of the long-term behaviors of all the qualitatively different solutions for the new variable, and then for the original equation.arrow_forwarda) A Multiple Linear Regression Model in the form provided in Equation 1 is required to predict the amount of trips likely to be generated in the Bortianor vicinity in 2040. Y=a+ajX1+æXz+æX3 (Equation 1) where Y= trips/household to work X1= number of cars owned by household X2 = income of household X3 = household size ao, ai, æ and a3 =constants Using the corelation matrix obtained from a base year survey in Table 5, detemine all the possible equations that can be fomulated in the fom of Equation 1 out of which the best predictive model will be selected. Give reasons for your choice of equations. Assume all the independent variables are linearly related to the dependent variable and eachindependent variable is easily projected. Table 5: Comrelation matrix Trips Household size Car Income ownership Trips Car ownership Income Household size 1 0.65 1 0.90 0.89 1 0.97 0.25 0.18 1arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning