Exercise 6.2.1 For each system of ODE in (a)-(g), do the following: • Formulate the system as x = Ax, by explicitly writing out the matrix A. • Find the eigenvalues and eigenvectors of A and use them to write out a general solution using (6.17). ● If any eigenvalues for A are complex, write out a real-valued general solution. • Use either form of the general solution to obtain the given initial data. (a) x₁ = 7x1 - 4x2, x2 = 20x1 - 11x2 with x₁ (0) = 3 and x₂ (0) = 8. (b) x1 = -x2, x2 = 6x15x2 with x₁ (0) = 2 and x2 (0) = 5. (c) x1 = x1 - x2, x2 = 5x1 - 3x2 with x₁ (0) = 0 and x₂ (0) = 2. (d) x₁ = -2x1 - 3x2, x2 = 3x12x2 with x₁ (0) = 2 and x₂(0) = -2. (e) x₁ = -6x1 +9x2 - 4x3, x2 = -6x1+11x2 - 6x3, x3 = -10x1 +21x2 - 12x3 with x₁ (0) = -1, x₂ (0) = 0, and x3 (0) = 2.