For the system of differential equations, 6 - 5 18 16 a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. A1, A2 = = U1 = 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. U2 = y' = y(t) d) Determine a general solution to the system. Enter your answer in the format y(t) = c₁f₁(t)v₁ + c₂f2(t)v₂. = C1 + c₂

Suppose the characteristic equation for a matrix A is given by X3 + 3X2 – 9A – 27 = 0. Find the eigenvalues of the matrix A and give the algebraic multiplicity of each eigenvalue. - Eigenvalue: A = 3 Eigenvalue: A = Algebraic multiplicity: Ex: 5 Algebraic multiplicity:

For the system of differential equations, 3 23 84 6 - 22 4et a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) A1, A2 b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. U₁ = U₂ = 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 λ₂. Enter the eigenvectors as a matrix with an appropriate size. y' = v(t) y + d) Determine a general solution to the system by completing the following steps. i. Find v(t) = [Y-¹(t)f(t)dt . ii. Find a particular solution y(t). H yp(t)= y(t) ii. Then a general solution for the system in the matrix form is -8:33:

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