dy5) Answer the following based on the differential equation = y² - y³.a) Find the critical points.b) Create a phase portrait. Sketch the solution in each region.c) Classify each critical point as stable, unstable, or semi-stable.d) Identify any equilibrium solutions.e) Solve the DE. Show work.

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Answered: 5) Answer the following based on the…

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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Consider the following differential equation. ÿ + 3y + 2y = 0 state space description ? b) After Solving for x (t) program to plot x(t) 'ist. =). Also plat x₂(t) so x₁ (+). d) Repeat. a), b) ¢ ¢) ific the differential equatio ÿ + w₂² y == o " and 2/0 = [b] (0) x(0) physical explanation to above results Give an write a computer - (x²)-[•])
Consider the following autonomous first-order differential equation. dy dx = (y − 2)4 Find the critical points and phase portrait of the given differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. (List the critical points according to their stability. Enter your answers as a comma-separated list. If there are no critical points in a certain category, enter NONE.) asymptotically stable unstable semi-stable
QI. Consider a spring mass system given in Fig 1 with mass of 1kg attached to a spring with K=10. The motion is damped with C=6 and is being driven by a force given by 25 cos 4t . If the spring is released from rest, find the equation of displacement y(t). Note: You cannot use the direct formulas for constants in particular solution. Spring Mass r(t) Dashpot Fig. 1
2. For a certain autonomous, first order differential equation with f as the dependent variable, you are told that there are 5 critical points (CP), located at -2 (an unstable CP), 0 (a stable CP), 2 (a semi-stable CP), 5 (an unstable CP), and 10 (a stable CP). a. Draw the appropriate Phase Portrait as a vertical line with critical points marked and arrows indicating direction of the rate of change. ( b. Sketch a graph of typical solution curve for values of y(0) above 10, between 10 and 5, between 5 and 2, between 2 and 0, between 0 and -2, and below -3 (including in the same sketch as your answer to part a.) t) c. If your initial value was f(0) = 15, what is the steady state solution for f d. What makes a Differential Equation autonomous?

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5) Answer the following based on the differential equation = y² — y³. a) Find the critical points. b) Create a phase portrait. Sketch the solution in each region. c) Classify each critical point as stable, unstable, or semi-stable. IL C *1'1