Given that v1 = [10] [65 -210] and v2 = are eigenvectors of the matrix 3 21 -68 determine the corresponding eigenvalues. λι = help (numbers) X2 = help (numbers) Find the solution to the linear system of differential equations x' = 65x-210y y' = 21x - 68y satisfying the initial conditions x (0) = 24 and y(0) = 7. x(t) = help (formulas) y(t) = help (formulas) Book: Section 3.4 of Notes on Diffy Qs

The matrix v₁ = 1 x₁ (t) = -13 12] -8 7 x₂ (t) = has eigenvalues A₁ = -1 and ₂ = -5. Find eigenvectors corresponding to these eigenvalues. satisfying the initial conditions 1 2 Find the solution to the linear system of differential equations and 7₂ x₁ (0)] _x₂ (0) = [3] help (formulas) help (formulas) = x 3 x = - 13 -8 12 7 X1 X help (matrices)

Consider the initial value problem -6 70) = -6 a. Find the eigenvalue A, an eigenvector 1, and a generalized eigenvector v, for the coefficient matrix of this linear system. 1 -6 1 b. Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in your answers. e^(-6t) (t+1)e^(-6t) ý(t) = c1 + c2 e^(-6t) c. Solve the original initial value problem. Y1(t) 2(t) = e^(-6t)

Suppose a system of differential equations yields eigenvalue λ=1+3i with 2+i eigenvector = Find any real-valued solution.

-57 26] The matrix has eigenvalues \1 = -5 and X2 = 8. Find eigenvectors -130 60 corresponding to these eigenvalues. v1 = [8] and v2 = help (matrices) Find the solution to the linear system of differential equations x' = -57x+26y satisfying the initial conditions x (0) y' = -130x + 60y = 1 and y(0) = 1. x(t) = = help (formulas) y(t) = help (formulas) Book: Section 3.4 of Notes on Diffy Qs

Find the eigenvenctors to this matrix

For the system of differential equations, = A₁, A₂ = U₁ = U2 = y' a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) = [16 b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. = -16 21 C1 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 X₂ . Enter the eigenvectors as a matrix with an appropriate size. 14 19 y d) Determine a general solution to the system. Enter your answer in the format y(t) = c₁f₁(t)ví + c₂f₂(t)v₂ . y(t) = + C₂

Using eigenvalues, find the general solution to the following system of equations. x' = x + y y' = -2x - y

A 3×3 real matrix has three eigenvalues. One of them is λ₁ = Another is A₂ = 1 + i with eigenvector v2 - (3) = -1 with eigenvector v₁ = (a) What is the third eigenvalue and its corresponding eigenvector? (b) Find the general solution to x' = Ax, in purely real form. (c) Find the solution to x' = Ax subject to initial condition x (0) = (:). (6) 1

Consider the linear system 3 *' = 27 a. Find the eigenvalues and eigenvectors for the coefficient matrix. help (numbers) help (matrices) A1 = , 01 = and help (numbers) help (matrices) 12 = , v2 = Find the real-valued solution to the initial value problem x = -3x1 – 2x2, x = 5x1+3x2, x1(0) = -4, x2(0) = 10. Use t as the independent variable in your answers. help (formulas) T2(t) help (formulas) OK Learn more Cookies help us deliver our services. By using our services, you agree to our use of cookies.