3. Consider the following system of equations, x(t) = (x, y), - x' = (3-1)x 5y y' = 2x+(3+ 1)y (a) Determine the eigenvalues of the system in terms of ẞ. (b) Determine the eigenvectors. (c) Find the general solution to the system in terms of ẞ. (d) Describe the different possible behaviours of solutions X (t) = (x, y) in the phase plane, for different values of the parameter B.

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.

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3. Consider the system of equations given by: a': -2 I. -3 a. Find a fundamental matrix for the system. x (t) = b. Find the matrix exponential, ¥(t) = e4", of the system. V(t) = (:) c. Solve the initial value problem with æ(0) = (. using the matrix exponential found in Part b. x(t) =

please solve it on paper

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

5c

Solve the system of differential equations. Find the lesser and greater of the two eigenvalues.

Given the system of equations, - G 1 5 х, х(0) — -3 x' use the fact that the eigenpairs of the system matrix are in order to answer the following. A = -4, uj = and d2 = 2, u2 (a) Give the general solution of the system of equations. (b) Solve the initial value problem.