Consider the following system of differential equationsdxdydtdtwith general solutionx= x + 2y,= α1e-t= -4x - 5y+02-3teSketch the phase portrait near the critical point at the origin. To sketch thephase portrait, determine:any special cases of the orbits,how the orbits behave as t→-∞,how the orbits behave as t→∞,the slope of the orbits when x = 0,the slope of the orbits when y = 0.In your sketch, show all the straight line orbits and at least four general orbits.

As per the question, we are given a system of differential equations with a general solution. Our…

Answered: Consider the following system of…

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 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.
Find a homogeneous linear differential equation with constant coefficients whose general solution is given. y = C₁ cos 5x + C₂ sin 5x Oy" + 25y = 0 Oy" - 25y = 0 Oy" - 10y' + 25y = 0 Oy" - 5y = 0 Oy" + 5y = 0
convert the given differential equation to a ﬁrst-order system using the substitution u = y,v = dy/dt , and determine the phase portrait for the resulting system
5. Sketch the phase line, classify the equilibrium points, and make a rough sketch of the graphs of the solutions for the following differential equations: dy dt = 3y(1-y)
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Consider the pair of differential equations dx dy (4 – x²) (y + 2), dt (x – 1) (3² – 1) . - dt (a) Find all the equilibrium points of these equations. (b) Using Maxima, or otherwise, plot the phase portrait of these equations. (c) Use your phase portrait to state the start and end of the phase path that goes through the origin.
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.
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(a) Consider the direction field of the differential equation dy/dx = x(y - 8)2 - 4, but do not use technology to obtain it. Describe the slopes of the lineal elements on the lines x = 0, y = 7, y = 8, and y = 9. X = 0 y = 7 y = 8 y = 9 (b) Consider the IVP dy/dx = x(y - 8)2 - 4, y(0) = Yor where y, < 8. Can a solution y(x) – o as x - o? Based on the information in part (a), discuss. O y(x) can approach o as x approaches o. O y(x) cannot approach co as x approaches o. Need Help? Read It Watch It
Consider the following autonomous first-order differential equation. dy dx = y2(25 − y2) 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 Sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.
Consider the following second-order partial differential equation : 1 yu - 2aryu.y + auyy ry (yu, + r*u,), > 0, y > 0. (G) 1. The equation (F) is : a) Hyperbolic Equation b) Parabolic Equation c) Elliptic Equation d) None of the above 2. The characteristics equations corresponding to (G) are: a) # = and = -2. b) = 3 and d = 4. %3D c) = d) None of the above. 3. Using the change of variable a = r² + y? and B = y, the canonical form of (G) is: a) u33 +ua = 0. b) uga +ua = 0. c) ugs -Jug = 0. d) None of the above. 4. Let f and g two arbitrary real valued functions. A general solution of (G) is: a) u(r, y) = f(r2 + y²) + y²g(x² + y?). b)u(r, y) = f(x) +ag(22 + y?). c) u(r, y) = f(y) +?g(a² + y?). d) None of the above.
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