Suppose that you are considering a system of differential equations: dx = ax + by dt dy = cx + dy dt x Suppose you know that one solution to the system is ·D=0 LY Consider the following two statements and decide whether they are true. Statement 1: a 6] The matrix must have eigenvalues 4 and 8. d Statement 2: = 8et 1 +4e5t 8 Select one alternative: © Statement 1 is true and statement 2 is true. Statement 1 is false and statement 2 is true. Statement 1 is true and statement 2 is false. © Statement 1 is false and statement 2 is false. c = et and another is must also be a solution to the system. [] 5 [8] = est

Express the given system of higher-order differential equations as a matrix system in normal form. x" + 5x+6y=0 y" - 5x=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? O A. x₁ = x¹, x₂ =y' O B. X₁ =X, X₂=X', X3 = Y, X4=y' OC. x₁ = x, X₂=x", X3 = Y. X4=y" OD. x₁ = x¹, x₂ = x¹, x₂ =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.) O A. O B. X₁ X₂ x₁² X1 x2 x3 X4 X1

A system of ordinary differential equations has 2 x 2 state matrix with eigenvalues and corresponding eigenvectors (eigenpairs) A1 = 5, v1 = and A2 = -6, Based on the given eigenpairs write down the general solution to the associated system of differential equations. The solution must be written as a |¤1(t) single vector like a =

Everything we've done with systems of 2 linear, constant co- efficient, homogeneous differential equations works for larger systems as well. Ask wolframalpha.com for the eigenvalues and eigenvectors of the following system of three differential equations, and use the result to write the general solution: -2 -4 2 *m-Biss X' (t) -2 1 2 X(t) 4 2 5 =