Consider the initial value problem:x1(0) = 4,x2(0) = 6.4x2,— 4т1 + 812,a. Find the eigenvalue A, an eigenvector v, and a generalized eigenvector w for the coefficient matrix of this linear system.b. Find the most general real-valued solution to the linear system of differential equations. Use c1 and c2 to denote arbitrary constants, and enter them as "c1" and "c2".x1(t) =x2(t)c. Solve the original initial value problem.x1(t)22(t)Instruction||||

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Answered: Consider the initial value problem:…

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 second-order equation d 2y/dt 2 + p dy/dt +qy = 0 a. If p=q=1, compute the eigenvalues and eigenvectors of the coefficient matrix. What would be the long-term behavior of the solutions in this case? b. If p=0 and q=2 determine the general solution.
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
Consider the second order differential equation as below:2?̈+ 12?̇ = 10?a) Change the above equation into matrix form of ?̇ = ??b) Write the quadratic form related to matrix A
Suppose A is a 2 × 2 real matrix with an eigenvalue λ = 3 + 5i and corresponding eigenvector i * = (-²+1). v= Determine a fundamental set (i.e., linearly independent set) of solutions for ÿ' = Aỹ, where the fundamental set consists entirely of real solutions. Enter your solutions below. Use t as the independent variable in your answers. (t) = 2(t) = → - →
1. Find the eigenvalues and the corresponding eigenvectors of the matrix A in the following system of linear equations 3x1 +x2 = Àx1 -5x1 – 3x2 = Ax2 %3D
13) The integer n = a) If n = 3 (mod 9), then what is x ? b) If n = 3 (mod 11), then what is x ? Hint for #13: 12700140x3243243 is missing the digit x. Consider the number 1234. By writing each digit in scientific notation, we have 1234 1 x 10³ +2× 10² +3 × 10¹ + 4 × 10⁰ Then notice that 10 = 1 (mod 9). When we reduce mod 9, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x 1³ + 2 x 1² + 3 × 1¹ + 4 × 1⁰ = 1+ 2+ 3+ 4 = 10 = 1 (mod 9). Since 10 = -1 (mod 11), when we reduce mod 11, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x (-1)³ +2× (-1)² +3 × (-1)¹ +4× (-1)⁰ = -1 + 2-3 + 4 = 2 (mod 11).
Suppose A is a 2 x 2 real matrix with an eigenvalue λ = 2 + 4i and corresponding eigenvector −1+ *-[-'+']. = Determine a fundamental set (i.e., linearly independent set) of solutions for y' = Ay, where the fundamental set consists entirely of real solutions. Enter your solutions below. Use t as the independent variable in your answers. y₁(t) = 19 y₂(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
Q3. (a) If Ax = 2 x, determine the eigenvalues and the corresponding eigenvectors for A= G ) 2 3. Hence, write down the associated modal matrix P and diagonal matrix D, and use these values to solve the following system differential equations: *1 = X1 + 4 x2 X2 = 2 x1 + 3 x2 Given that when t = 0, x1 = 0 and x2 = 2. (b) Use the 4th order Runge Kutta method to solve the differential equation: dy e* y dx for values of x 0 (0.2) 0.4 given that y 1 when x 0. %3D Give your answers correct to 5 decimal places. Obtain the analytical solution of the differential equation and compare the analytical solution when x = 0.4 with the values obtained using Runge Kutta,
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.
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 =
Consider the following matrix A = 5 7 0 -2 (a) Find the eigenvalues and the eigenvectors of this matrix. (You need to show how you find them.) (b) Using the eigenvalues and the eigenvectors of A, solve the system of differential equations ' = AT.

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