(a) Express the following inhomogeneous system of first-order differentialequations for r(t) and y(t) in matrix form:* = 2x + y + 3e",y = 4x - y.eWrite down, also in matrix form, the corresponding homogeneoussystem of equations.Pod(b)Find the eigenvalues of the matrix of coefficients and an eigenvectorcorresponding to each eigenvalue.(c) Hence write down the complementary function for the system ofequations.(d) Find a particular integral for the original inhomogeneous system.(e) Hence write down the general solution of the original inhomogeneoussystem.(f) Find the particular solution of the original inhomogeneous system withx = 3 and y=-3 when t = 0.(g)What is the long-term behaviour of this particular solution as t becomeslarge? Does the ratio y/x tend to a fixed number, and if so whatnumber?

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Answered: Express the following inhomogeneous…

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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Solve the system of differential equations given in matrix form for the general solution. We must do this by calculating the eigenvalues and eigenvectors. Note: we get a repeated eigenvalue and must solve for the second eigenvector to get two linearly independent solutions for our general solution. Please explain how you calculate the second eigenvector
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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
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Consider the system of differential equations = (-3/2 3/2)x+ (-¹1) ¹ x' = -3/2 -1 where two eigenvalues of 2 3/2 -3/2 -1 are r = 1/2 and r₂ = 1/2, and the corresponding eigenvector is v= -(2/3). Let x = Ty where T is a transformation matrix. Find two first order differential equations, eigenvector is u=== and = -/+nte, n =+², ₂ -+-+ M and the generalized
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For the system of differential equations, X1, X2 a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. U₁ = U2 = = y' = c) Find the eigenvectors assuming u₁ is the eigenvector associated with the smaller eigenvalue X₁ and u2 is the eigenvector associated with the larger eigenvalue A2. 5 2 - 1 Y2 (t) d) Determine two linearly independent solutions to the system. Enter the first solution in the format y₁ (t) y = = y₁ (t) Enter the second solution in the format y₂(t) = ƒ2(t) (v3h₁(t) + v₁h₂(t)) = : fi(t) (vigi(t) – v2g2(t)) . +
Everything we've done with systems of 2 linear, constant co- efficient, homogeneous differential equations works for larger systems as well. Ask wolframalpha.com for the eigenvalues and eigenvectors of the following system of three differential equations, and use the result to write the general solution: -2 -4 2 *m-Biss X' (t) -2 1 2 X(t) 4 2 5 =

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