Consider a system of two nonlinear first-order ODEs, where x and y are functions of theindependent variable t:1* = 2tanh(x) — 2x cos(y) + e²+3y − 1, y = 3 cosh(x) − 3e²y + y + =sin(x). (a) Write down in matrix form of the type X = AX with X = (x, y) the systemobtained by linearisation of the above equations around the point x = y = 0.Specify the elements of the matrix A.(b) Find the eigenvalues and eigenvectors of the matrix A obtained in (a). Writedown the general solution of the linear system.

As per the question we are given a nonlinear dynamic system and we have to : Linearize the system…

Answered: Write down in matrix form of the type X…

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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Express the given system of higher-order differential equations as a matrix system in normal form. x" + 5x+6y=0 y" - 5x=0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? O A. x₁ = x¹, x₂ =y' O B. X₁ =X, X₂=X', X3 = Y, X4=y' OC. x₁ = x, X₂=x", X3 = Y. X4=y" OD. x₁ = x¹, x₂ = x¹, x₂ =y', X4=y" Write the system of equations using matrix notation. Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) O A. O B. X₁ X₂ x₁² X1 x2 x3 X4 X1
The coefficient matrix A of the 4x4 system below has eigenvalues A₁-9, A₂-6, Ag 9, and 4 14. Find the particular solution of this system that satisfies the initial conditions x₁ (0) = 5, x₂ (0) = 5, x₁ (0) = 5, x4 (0) = 5. x₁ = 2x₁ + x₂ + xy + 8x4 x2 = x₁ + 2x₂ + 11xg+X4 X3 X₁ +11x₂ + 2xy +Xg x48x₁ + x₂ + xy + 2x₁ *(t) =
I tried -210e^-7t but it didn't work and idk what i did wrong...
X' = {(1, 2), (4, 3)} x X + {(8e3t), (12e2t)}. Where X1 = {(-e-t), (e-t)} and X2 = {(e5t), (2e5t)} are solutions of the corresponding homogeneous system. Show that the solutions of the corresponding homogeneous system are linearly independent. Find the fundamental matrix and its inverse. Solve the nonhomogeneous system given above. Show all of your work and multiply out all matrix calculations.
Need only a handwritten solution only (not a typed one).
Suppose the eigenvalues associated with a 2 by 2 matrix of a system of linear ODE's are complex conjugates. (a) Use Euler's formula to rewrite the solution in the form x(t) = u(t) + iv(t). (b) Show that x(t) = u(t) + v(t) is also a solution of the system.
f) Use your eigenvalues and eigenvectors to write the general solution, in matrix form, to your linear system from part (a). (Your answer should have two arbitrary constants). g) Apply your initial conditions from part (a) to solve for the arbitrary constants. Please answer a-h with clear steps thanks h) Take your solution out of matrix form and rewrite it in the form Q1= f(t), Q2 = g(t)
The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system. x', = 4x, + 2x2 + 2x3, x'2 = - 6x, - 4x2 - 2x3, x'3 = 6x, + 6x2 + 4x3 What is the general solution in matrix form? x(t) =
The coefficient matrix has a zero eigenvalue. As a result, the pattern of trajectories is different from those in the examples in the text. For each system: Find the general solution of the given system of equations.

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