Apply the eigenvalue method to find a general solution of the given system. Use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.x', = - 2x, + 8x2. x'2 = 12x, - 6x2What is the general solution in matrix form?x(t) =O

Solution of the system of linear differential equations

Answered: Apply the eigenvalue method to find a…

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 system of differential equations in matrix notation. x' = 3x +9y y' = 4x-9y 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.) tor OA. OB. OC.
Find the eigenvalues of the coefficient matrix for the system.
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By using the method of least squares, find the best parabola through the points: (-1,-3). (1,3). (-2,-3). (0,-3) Step 1. The general equation of a parabola is c₂+₁+c₂2²-3. Plugging the data points into this formula gives a matrix equation Ac=y. 1 ATA- ATy= 1 Step 3. Solving the normal equation gives the answer -8 which corresponds to the formula a Step 2. The matrix equation Ac=y has no solution, so instead we use the normal equation A¹A = A¹y O 0 3
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A=30 ,B=20, C = |,D= 22 Find the 3x3 homogeneous matrix for 2D-reflection across a line that goes through (B, C) and makes an angle at +D degrees (counterclockwise) from the x-axis.
b) A system of differential equation is given as follows: x₁ = 9x₁ - 4x₂ x₂ = 8x₁ - 3x₂ where x₁ and X2 are functions of . Given that the eigenvalues of A= and 5 with the corresponding eigenvectors i) Write a matrix P that diagonalizes A. ii) Write a diagonal matrix D such that P¹ AP = D. iii) Hence, find the general solution to the system. [4=[83] are 1 and respectively.
dx dt = x+y+z; (a) (b) dy dt = x + 3y - z; dz dt = 2y + 2z For the system above, define matrix A and vector X and write the system in matrix form. Show all work to solve the system and clearly state the general solution vector X
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Apply the eigenvalue method to find a general solution of the given system. Use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. x₁ = 9x₁ - 10x₂, x 2 = 8x₁ + x₂ What is the general solution in matrix form? x(t) = F More e
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.
Find the equation satisfied by the squares of the singular values of the matrix associated with the following over-determined set of equations: 2x + 3y + z = 0 x- y – z = 1 2x + y = 0 2y + z = -2. Show that one of the singular values is close to zero. Determine the two larger singular values by an appropriate iteration process and the smallest one by indirect calculation.

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Apply the eigenvalue method to find a general solution of the given system. Use a computer system or graphing calculator to construct a direction field and typicai solution curves for the given system. x', = - 2x, + 8x2. x'2 = 12x, - 6x2 What is the general solution in matrix form? x(t) = D