[17] For each of the following equations,• find general solutions;• solve the initial value problem with initial condition y(0) = –1, y'(0) = 2;• sketch the phase portrait, identify the type of each equilibrium, and determine thestability of each equilibrium.(a) 2y" + 9y' + 4y = 0(d) 2y" – 3y = 0(g) 9y" + 6y' + y = 0(b) у" + 2у - 8у —D0(е) у"- 2у + 5у 3D0(c) 4y" – 12y + 5y = 0(f) 4y" + 9y = 0-

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Answered: [17] For each of the following…

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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[11] For each of the following equations, find general solutions; solve the initial value problem with initial condition x₁ (0) = -1, x₂ (0) = 2; • sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. -2 -13 dx₁/dt = x₁ + x2 x₁ = x₁4x₂ (a) x' = [ X (b) { 4 dx2/dt x₂ = 5x₁²x₂ 1 = -2x1 - x2
[14] For each of the following equations, find all equilibria; • find general solutions; • solve the initial value problem with initial condition ₁ (0) = 2, x₂(0) = 1; • sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. 10 x1 1 1 - 3-4 963 = (b) Q]=[4][B]+[B] x2 -5 -7 3 (a)
[17] For each of the following equations, • find general solutions; • solve the initial value problem with initial condition y(0) = –1, y'(0) = 2; • sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. (a) 2y" + 9y' + 4y = 0 (b) у" + 2у - 8у —D0 (с) 4" — 12у + 5у — 0
[6] An equation dt = f(y) has the following phase portrait. 2 Y (a) Find all equilibrium solutions. (b) Determine whether each of the equilibrium solutions is stable, asymptotically stable or unstable. (c) Graph the solutions y(t) vs t, for the initial values y(1.4) = 0, y(0) = 0.5, y(0) = 1, y (0) = 1.1, y(0) = 1.5, y(-0.5) = 1.5, y(0) = 2, y(0) = 2.5, y(0) = 3, y(0) = 3.5, y(0) = 4, y(0) = 4.5, y(-1) = 4.5. (Without further quantitative information about the equation and the solution formula, it's clearly impossible to draw accurate graphs of y(t) vs t. Here, try to sketch graphs qualitatively to show the correct dynamic properties. The point is that a great deal of info about solution dynamics can be read off from one simple figure of phase portrait.)
For the following problems: ii. Identify the equilibrium values Construct a phase line. Identify the signs of y' and y". Sketch several solution curves. a. = (y + 2)(y - 3) dx b. = y²-2y dx
[17] For each of the following equations, • find general solutions; • solve the initial value problem with initial condition y(0) = -1, y'(0) = 2; sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. (a) 2y" + 9y' + 4y = 0 (d) 2y" – 3y' = 0 (g) 9y" + 6y' + Y = 0 (b) y" + 2y' – 8y = 0 (e) y" – 2y' + 5y = 0 (c) 4y" – 12y' + 5y = 0 (f) 4y" + 9y = 0
2. Analyse the following equations graphically: x = ex COS X x = 1- cos 2x (ii) Find all the equilibrium points, classify their stability and sketch the graph of x(t) for different initial data.
[13] For each of the following equations, find general solutions; • solve the initial value problem with initial condition x₁(0) = -1, x₂ (0) = 2; • sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. x₁ +4x₂ -5 0 (a) =[ 1 d x1 dt X2 () = »E]-[73][B] ©*-[72]* (c) = 4 -x1 + 5x₂ - X2 0
For the Attached problem, Identify the equilibrium values. Which are stable and which are unstable? Construct a phase line. Identify the signs of and Sketch several solution curves
3.as soon as possible please
1) Find all equilibrium solutions of the equation (1 − x) (x² − 4) - x = and classify each one in terms of stability. Draw a phase space diagram and sketch by hand several typical solution curves. Describe the long term (t → ±∞) behavior of the solutions.
7. For each of the following equations sketch the phase portrait of the correspond- ing first order system. Then sketch the graphs of several solutions s(t) for different initial conditions: (b) s". (c) s" + s'+s=0 (a) s" + s : 0 = -8= 0

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