Consider the initial value problem i' = y, j(0) = a. Find the eigenvalue A, an eigenvector i1, and a generalized eigenvector v, for the coefficient matrix of this linear system. 1 1 A= -6 v1 = ,V2 = 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. ý(t) = C1 c. Solve the original initial value problem. Y1 (t) = || Y2(t)

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You are solving a 2x2 system of the form : y' = PỶ + where P has constant entries with real different eigenvalues. Decide which of the following statements is correct: I can find a particular solution using the undetermined coefficients method with a try function of the form Y(t) = sin(t)å + cos(t)b I can find a particular solution using the undetermined coefficients method with a try function of the form Y(t) = (cos(t) + sin(t))ā I can only find the general solution for this problems using the Variation of Parameters method. Undetermined Coefficients won't work in this case. I can find the general solution using the undetermined coefficients method. To find the particular solution I should use a with a try function of the form Y(t) = sin(t)ä

Consider the system of ODEs: x' = -3x – y y' = 4x + 2y Find the equilibrium solution(s) and classify them based on the eigenvalues and eigenvectors of the associated coefficient matrix. You do NOT need to solve the system. Please type in your step-by-step solution.

Express the given system of higher-order differential equations as a matrix system in normal form. x" + 6x + 7y=0 y" - 2x=0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? OA. X₁ = x¹, x₂ = y' O B. X₁ =X, X2₂=X'', X3 = Y, X4 =y" O c. x₁ = x¹, x₂ = x¹, X3 = y', X4 =y" O D. x₁ = x, X₂ = x', X3 = Y, X4 = 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.) OA. B. X₁ X₂ X₁ S w X4 || X₁ X2 x1 3 ... W X4 ро

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) = → - →

) Consider the linear system X₁ = Y a. Find the eigenvalues and eigenvectors for the coefficient matrix. 18 v1 = b. Find the real-valued solution to the initial value problem {{{ = -3 -2 5 -3] Use t as the independent variable in your answers. y₁ (t) Y₂ (t) - 3y1 - 2y2, 5y1 + 3y2, y." , and X₂ = V2 y₁ (0) = 9, Y₂ (0) = -10. ||

Consider the system of equations vec(x′) = Ax, where A is a 2×2 matrix. If λ=2+i is an eigenvalue of A with corresponding eigenvector vec{v} = (1, −1−i​), express the solution to the system of equations in terms of real-valued functions.

Solve the following initial value problems for the systems of equations using the matrix method. Find eigenvalues and eigenvectors by hand (but you can use technology to check your answers) (a) (b) x' = y' I x2 - 6x + 3y -4x-y 2 2x1 - 5x2 4x1 - 2x2 x(0) = 3, y(0) = -2. 2₁ (0) = 1, ₂ (0) = 1. 7

Consider the initial value problem 4x2, x1(0) = 2, -4x1 + 8x2, x2(0) = 6. Find the eigenvalue A, an eigenvector v1, and a generalized eigenvector v2 for the coefficient matrix of this linear system. 1 4,4 1 1 help (numbers) help (matrices) Find the most general real-valued solution to the linear system of differential equations. Use c and c2 to denote arbitrary constants, and enter them as "c1" and "c2". x1(t) cle^(4t) help (formulas) x2 (t) help (formulas) Solve the original initial value problem. x1(t) : 4e^(4t) help (formulas) x2(t) : 4e^(4t)+2 help (formulas)