The next problem involves an equation of the dy/dt = f(y). For the equation given determine the critical(equilibrium) points, classify each one as asymptotically stable or unstable. Draw the phase line, and sketchseveral graphs of solutions in the ty-plane (be sure to check concavity in this part).dydt= y(y − 1)(y — 2),dydt= ey - 1,Yo ≥ 0.-∞< Yo <∞.

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Answered: The next problem involves an equation…

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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dy 3. Use the phase plot for dt = g(y) to identify the equilibria (if any) and their corresponding stabilities. 1.5- 1- 0.5- 0.5 1.5 2.5 3.5 -0.5- Equilibria Stability
Q4: Sketch the phase plane portrait of equation (without explicitly computing its solutions), x"(t) - 2x' + 2x = 0. What type is it, what happens to solutions as t → ∞o?
The control system which stabilises the flight dynamics of the jet fighter is given by the 2nd order differential equation: d2y/dx2 - 2(dy/dx) - 8y = 0 Consider the particular solution with Initial Conditions: y(0) = 0.5 and y'(0) = 4. Solve for the vertical motion (y) using Laplace and Inverse Laplace transforms.
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).
Solve B16b. and perform four iterations.
Solve the following initial-value problem, determine the type of the equilibrium -ion (0,0), and sketch a phase portrait: dx x + y, x (0) = 0 dt dy 4x + Y, y (0) = 3 dt
Revert Don. 6. The simulation of parameter-distributed processes is connected with discretization in space and time. The distribution of changes in the temperature x of a heated at the front massive long metal piece is described by the following partial differential equation: a²x(z,t) Oz² ax(z,t) (1) = -a(x-0₂)+b at where 0, is the ambient temperature, a and b are constants. Derive the discrete model by applying discretization first with respect to z (z;= i.Az) and after that with respect to t (tk=k.At), using backward finite differences for the corresponding derivatives.
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.
b T4
2a. 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.
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
Construct an autonomous first-order differential equation dy/dx = f(y) whose phase portrait is consistent with the given figure.

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