8. Find a vector b such that the system- (¹-1/²-3/2)x+bthat could have the phase portrait shown below with a critical point at (1,1)21.50.50-0.5-1x'=-0.500.5X11.52

Linear stability analysis:Let X'=JX be a linearized system at the critical point, then If the…

Answered: 8. Find a vector b such that the system…

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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For the equation dy/dt=y^4+4y^3-5y^2 (a) Find all the equilibrium solutions and determine their stability type (b) sketch the phase line and draw several solutions curves in the ty plane (c) determine all the inflection levels, if any at all, for the solutions curves
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This is the first part of a two-part problem. Let P=[-: 1 5₁(t) = [(41) 5₂(t) = - sin(4t). a. Show that y₁ (t) is a solution to the system ÿ' = Pÿ by evaluating derivatives and the matrix product y(t) = = 0 [1] -4 Enter your answers in terms of the variable t. -4 sin(4t) -4 cos(4t)] ÿ₁ (t) [181-18] b. Show that y₂ (t) is a solution to the system ÿ' = Pÿ by evaluating derivatives and the matrix product Enter your answers in terms of the variable t. 04] 32(t) = [-28]|2(t) 181-181
The vibrating system governed by + x + x = 0 has an initial position x(0) = 4 and an initial velocity (0) = 0. Write down an analytic expression for x(t).
This is the first part of a two-part problem. Let O 21 P = -2 sin(2t)] -2 cos(2t)]* cos(2t) y1(t) sin(2t)| ÿ2(t) = a. Show that 1(t) is a solution to the system i' = Pý by evaluating derivatives and the matrix product O 2] -2 Enter your answers in terms of the variable t. b. Show that y2(t) is a solution to the system i' = Pj by evaluating derivatives and the matrix product %(t) y2(t) Enter your answers in terms of the variable t.
Exercise: Solve the following system x'=y +1 y'=x +1 (a)] (b)
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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.
Investigate the type of critical point (0,0) of the given linear system. dx/dt = -6x+13y+x^3 dy/dt = -x-2y+y^4

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8. Find a vector b such that the system (1/2 x = (¹/² - 3/2)x+b that could have the phase portrait shown below with a critical point at (1, 1)