5.Consider the following autonomous first-order differential equation.dyda= y²(4- y²)(a) Which of the following is the phase portrait for the differential equation?0-2B(b) List the critical points and classify each critical point as asymptotically stable, unstable, orsemistable.(c) Sketch the actual solutions in the r- y plane. Make sure to sketch the equilibrium solutions.Make sure you sketch typical solution curves in all the regions in the a-y. Make sure to labelyour graph.Y46-2420-02

Answered: 5. Consider the following autonomous…

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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Sketch the phase picture associated with (3) in the (q, p) plane. Note the equilibrium points and other special lanes.I have the solution, I just dont know how they get there, so please if able explain the steps in detail to get there, I am very new to differential equations, thank you in advance. the equilibrium points: E0 = (0, 0), E1 = (1, 1), E2 = (1, −1), E3 = (−1, −1), E4 = (−1, 1).
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
5
4. Consider the autonomous ODE, y' = f(y) = -y(y² – y – 6). (a) Identify the equilibrium solutions and classify the stability of each. d²y directly dt? (b) Identify points of inflection for the solution curves by finding an expression for from the autonomous differential equation. d to the equation dt dy = f(y).) dt (Apply - (c) Sketch a few solution curves in the (t, y) plane, including the location of inflection points.
dy dx 2. Construct a direction field for the differential equation = 3y(1y). Then sketch a few solution curves for different initial conditions, making sure to include the solution curves with initial conditions y(0) = 0 and y(0) = 1 plus a least one solution curve above and below each horizontal solution curve (which are called equilibrium solutions). Y X
2. [15 points] The following differential equation describes the steady-state concentration of a substance that reacts with first-order kinetics in an axially-dispersed plug-flow reactor, as shown in Figure 1: d² c D dx² dc dx U kc = 0, where D = the dispersion coefficient (m2/h), c = concentration (mol/L), U = the velocity (m/h), and k = the reaction rate (1/h). The boundary conditions can be formulated as Ucin = Uc(x = 0) - D dc (x 0) dx dc (x = L) = 0, dx where Cin = the concentration in the inflow (mol/L), and L = the length of the reactor (m). Use the finite-difference method to solve for concentration as a function of distance, given the following parameters: D = 5000 m²/h, U = 100 m/h, k = 2 1/h, L = 100 m, and cin = 100 mol/L. Use centered finite-difference approximations with Ax = 10 m to obtain your solution. Compare your numerical results with the analytical solution: C= Ucin (U — DÀ1)À2e^2ª — (U — DÀ2)À¹³1 × (+20 +212 - 1+1+2) - 4kD ---/ (+1√17) - 10 (1√+420) U 1 = 2D 4kD U…
Given the DE: y"-y'-12y=0 y"+y'+16y=0 2y"+8y'+6y=0 a. Convert the second-order differential equation into a system of two first-order differential equations. b. Plot the phase plane for the system. c. Describe how y changes as time goes on.
#2
Consider the following autonomous first-order differential equation. dy/dx = y2(36 − y2)
Construct an autonomous first-order differential equation dy/dx = f(y) whose phase portrait is consistent with the given figure.
Note k = 3. differential equation

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