a. Find the eigenvalues and eigenvectors of the matrixy₁ (t) =y₂(t) =λ₁[y₁ (0)]L32 (0)=21 24]-16 -192124]b. Solve the system of differential equations Y' = [--16 -19satisfying the initial conditions- [1]=V₁ =[8]1Yand X2=V₂ = =181

Answered: Image /qna-images/answer/662c1bb8-f269-4ba2-9983-cf2353063c6d.jpg

Answered: a. Find the eigenvalues and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

4. A real, symmetric 3-by-3 matrix M has eigenvalues X₁ = 4, X₂ 1 vector corresponding to the eigenvalue X₁ = 4 is v₁ -1 0 [4] -2 to the eigenvalue X₂ = 6 is v₂ = 6 and X3 = -3. An eigen- and an eigenvector corresponding What is M?
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
12- The system x = x + 4y, y' = y + x has two eigenvectors....
35. (a) Show that the eigenvalues of the 2 X 2 matrix a b A = c d] с а are the solutions of the quadratic equation 12 – tr(A)A + det A = 0, where tr(A) is the trace of A. (See page 162.) (b) Show that the eigenvalues of the matrix A in part (a) are A = {(a + d ± V(a – d)? + 4bc) (c) Show that the trace and determinant of the matrix A in part (a) are given by tr(A) - λι + λ and det A = λιλε where A, and A2 are the eigenvalues of A. 36. Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has (a) two distinct real eigenvalues, (b) one real eigenvalue, and (c) no real eigenvalues.
15. Find the eigenvalues and cigenvectors of matrix A. Then find the general solution to X(t + 1) = AX(t). (a) A = 1 2 () 2 () 0 3 0 -3 (b) A = (43)
For (a) and (b), 2 -1 [A] = -1 2 (a) What are the eigenvalues of A? (b) For each eigenvalue, find a nonzero eigenvector. [1/3] [1/4] (c) Suppose that the matrix B has the eigenvalues 1 and -6, with eigenvectors and respectively. Calculate Q(t).N(0)'. (Q: fundamental matrix for u’=B.u) (d) What is the solution to u'=B.u with u(0)=| ?
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) = → - →
Let A be a 2 x 2 matrix with eigenvalues X₁ = -3 and >₂ = -1 and [1] an H Let x(t) be the position of a particle at time t. Suppose we solved the initial corresponding eigenvectors V₁ = value problem x = Ax, x(0) = [3] -ce + c₂e-t x(t) = = [2(t)] = [ qe * +ge + -3t Search and C2= -3t and V2 = to obtain then C1=
How large should n be to guarantee that the Trapezoidal Rule approximation to x - 12x- 48x² + 2x + 5) dæ is accurate to within 0.1. n = How large should n be to guarantee that the Simpsons Rule approximation to 2 1(-a - 12a - 48z? + 2x + 5) dx is accurate to within 0.1. n = Hint: Remember your answers should be a whole numbers, and Simpson's Rule requires even values for n Question Help: Video 1 D Video 2 Submit Question P Type here to search hp
Suppose that a 2 x 2 matrix A with real entries has an eigenvalue X = 1 + 3i with corresponding 4 i [12] eigenvector 1 + 2i Which of the following is NOT a solution of the linear system y' = Ay? y(t) = et y(t) = et y(t) = et y(t) = et 4 cos(3t) sin(3t) [cos(3t) + 2 sin(3t). 4 cos(3t) + sin(3t) cos(3t) 2 sin (3t) - cos(3t) + 4 sin(3t) 2 cos(3t) + sin(3t) cos (3t) - 4 sin (3t) -2 cos(3t) - sin(3t)
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,
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.

SEE MORE QUESTIONS

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

a. Find the eigenvalues and eigenvectors of the matrix y₁ (t) = y₂(t) = λ₁ [y₁ (0)] L32 (0) = 21 24] -16 -19 21 24] b. Solve the system of differential equations Y' = [- -16 -19 satisfying the initial conditions - [1] = V₁ = [8] 1 Y and X2 = -181 V₂ = =