Consider the initial value problem'=[¯6 __!];, ;(0) =(0) = [₂]a. Find the eigenvalue A, an eigenvector ₁, and a generalized eigenvector ₂ for the coefficient matrix of this linear system.λ =.v₁ =-1b. Find the most general real-valued solution to the linear system of differential equations. Use I as the independent variable in your answers.y(t) = C₁+0₂c. Solve the original initial value problem.y₁ (1) =y₂ (1) =

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Answered: ' - [6 __!]. ÿ() = [] Find the…

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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Consider the Initial Value Problem: x₁ x2 X₁ = 0,₁ % v1 = 2x1 + 2x2 = = -4x12x₂² x1 (0) = 4 x2 (0) 6 (a) Find the eigenvalues and eigenvectors for the coefficient matrix. 181 = , and X₂ = ₁0₂ = (b) Solve the initial value problem. Give your solution in real form. x1 x2 = An ellipse with clockwise orientation phase plotter pplane9.m in MATLAB to describe the trajectory. [B] 1. Use the
A linear systems of differential equations is to be solved using eigenvalue method. Determine the eigenvalues of the system and obtain corresponding eigenvectors to get the general solution. (Hint: You may need to define a new variable. ) y' = x x" = 3x + 2y
Consider the initial value problem A₁ = x(t) = v1 = dx (a) Find the eigenvalues and eigenvectors for the coefficient matrix. 18) = dt = x, x(0) = , and X₂ = 72 = (b) Solve the initial value problem. Give your solution in real form. 18 (181
Consider the following initial-value problem. (;小) -1 -2 X' = X(0) = + (:)- Find the eigenvalues of the coefficient matrix A(t). (Enter your answers as a comma-separated list.) = Find an eigenvector for the corresponding eigenvalues. (Enter your answers from smallest eigenvalue to largest eigenvalue. K1 K, = Solve the initial-value problem. X(t) =
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)) . +
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) =
If A is an (n x n)-matrix of real constants that has a complex eigenvalue and eigenvector v, then the real and imaginary parts of w(t) = elty are linearly independent real-valued solutions of x' =Ax: x₁(t) = Re(w(t)) and X₂(t) = Im(w(t)), The matrix in the following system has complex eigenvalues; use the above theorem to find the general (real-valued) solution. 0 30 -3 0 0 X 005 x(t) = = x' = Find the particular solution given the initial conditions. x(t) = = x(0) = 123
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) =
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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 =
Solve the given system of differential equations by systematic elimination. (2D2 D-1)x - (2D + 1)y = 8 (D1)x+ Dy = -8 (x(t), y(t)) = = Need Help? Read It Watch It

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