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.

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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) =

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

Express the system of differential equations in matrix notation. x' = (e ¹)x+ (at + bt³)y y' = (tan t)x+ (csct)y Write the system of equations using matrix notation. Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) O A. OB. O C. X' y' = X y [-] 0-40

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.

Consider the linear system X1 = and Find the eigenvalues and eigenvectors for the coefficient matrix. A₂ = 0 JI ← A, V₁ 9 V2 = 81 = = -3 6 - - 4 x. help (numbers) help (matrices) help (numbers) help (matrices) For each eigenpair in the previous part, form a solution of a' = Ax. Use t as the independent variable in your answers.

O A₁ = 2, V₁ = OA₁2, V₁ = O A₁ = 2, V₁ = The phase portrait shown above is for a linear system associated eigenvectors: O X₁ = -2, V₁ 5 [3] M [³] 5 3 = 0 A₁ = -2, V₁ = = [3] 5 fre frame [3] 5 4 and X₂ = 3, V₂ = = and A₂-3, V₂= = and A₂3, V₂ = and A₂3, V₂ = dy dt and A₂-3, V₂ = 0 H H H = = 0 7 -N 7 1 1 1 3 x AY where the matrix A has eigenvalues and

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Consider the linear system 3 x = x. Find the eigenvalues and eigenvectors for the coefficient matrix. A₁ = 0, 0₁ = v1 λι and X2 ×2 = 0,02 = [8] [8] help (numbers) help (matrices) help (numbers) help (matrices) For each eigenpair in the previous part, form a solution of ' independent variable in your answers. = Ax. Use t as the x1(t) = and x2(t) = [8] [8] help (formulas) help (matrices) help (formulas) help (matrices) Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solutions? Choose Book: Section 3.4 of Notes on Diffy Qs