4. Consider the linear system of differential equationsa-1(H-G 200)=1a(1)where a € IR is a parameter.(a) Find the eigenvalues and the corresponding eigenvectors of the matrix of thesystem, and write the general solution of system (1).(b) Characterise the steady state (0,0) as a varies from -∞ to +∞0.(c) Sketch the phase portrait of the system as accurately as possible for thefollowing values of the parameter a. In each case explain your reasoning.(i) a = -1, (ii) a = 0.

Eigenvalue: A number is said to be an eigenvalue of the matrix A if there exists a non-zerovector x…

Answered: 4. Consider the linear 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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1. (a) What are A, X and x' in matrix form x' = AX of the system of differential equations { FO (b) Find all eigenvalues and eigenvectors of A. (c) Find exponential solutions of the system of differential equations. (d) Verify your solutions. x₁ = 3x1 x2 = 5x1 + 2? x2
The behaviour of an unforced mechanical system is governed by the differential equation 5 2 -1 0 x(1) = 3 6-9 x(t), x(0) = 1 1 1 1 0 (a) Show that the eigenvalues of the system matrix are 6, 3, 3 and that there is only one linearly independent eigenvector corresponding to the eigenvalue 3. Obtain the eigenvectors corresponding to the eigenvalues 6 and 3 and a further generalized eigenvector for the eigenvalue 3. (b) Write down a generalized modal matrix M and confirm that AM = MJ for an appropriate Jordan matrix J. x(t) = Me M-¹x(0) obtain the solution to the given differential equation. (c) Using the result
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
Express the given system of higher-order differential equations as a matrix system in normal form. x" + 6x + 7y=0 y" - 2x=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? OA. X₁ = x¹, x₂ = y' O B. X₁ =X, X2₂=X'', X3 = Y, X4 =y" O c. x₁ = x¹, x₂ = x¹, X3 = y', X4 =y" O D. x₁ = x, X₂ = x', X3 = 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.) OA. B. X₁ X₂ X₁ S w X4 || X₁ X2 x1 3 ... W X4 ро
Consider a system of differential equations which has the general solution: 18]- = C, · cos (5. t) – sin (5 - t) )+C2 · sin (5 - t) + • Cos (5 - t (a) Determine the fundamental matrix (t) which has the property (0) = I. (t) = (b) Use the fundamental matrix found in part (a) to quickly determine the particular solution for the following initial conditions: r (t) : y (t) = :8]-[- (0): 1 (ii) r (t) = y (t) =
Suppose a system of first-order linear differential equations has eigenvalues A₁ = -3, A2=-2.3. The equilibrium point (origin) can be classified as: O Flux Point O Saddle Point O Origin Point O Stable Node (Sink) O Unstable Node (Source)
5- Find the differential equation for the system below. x(t) y(t) a) O x(t) = y "(t) + (a,+a2)y (t) b) x(t) = y "(t) + (a,+a2)y '(t) + a2 C) none of these d) x(t) = y "(t) + (a1 - a2)y (t) e) x(t) +y "(t) + (a+a2)y (t) + a2 =0 f) x(t) = y "(t) - (a + a2)y (t)
A system of ordinary differential equations has an equilibrium point at the origin with the Jacobian matrix J = [0 1][-3 0] what is the classification of the linearised system of equations at this equilibrium point? - Sink, centre, spiral source or star source?
Ок Denote the owl and wood rat populations at time k by x = Rk where k is in months, Ok is the number of owls, and R is the number of rats (in thousands). Suppose OK and RK satisfy the equations below. Determine the evolution of the dynamical system. (Give a formula for XK.) As time passes, what happens to the sizes of the owl and wood rat populations? The system tends toward what is sometimes called an unstable equilibrium. What might happen to the system if some aspect of the model (such as birth rates or the predation rate) were to change slightly? Ok+1 = (0.2)0k + (0.5)RK Rk+1=(-0.16)0k + (1.1)Rk Give a formula for xk- xk = 4 ( D +C₂1

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