Hello, I have differential equation questions.

I need your help with explanation.

Thank you so much.

1. Consider the system x' = Ax, where 3 7 A _ [ 1 _7 l . (a) Find the eigenvalues of A. (b) Find the associated eigenvectors. (c) Find two linearly independent solutions, and show that they are linearly indepen- dent by calculating the Wronskian. E. (d) Find the value of 231(2". ) if x0 = [g] . 2. Consider the system x' = Ax + g(t), where 6 1 1 t A: 1 4 1 and g: t 0 0 5 1 (a) For the corresponding homogeneous system, nd the special fundamental matrix, (t), such that @(0) = I. (b) Find _1(t) (recall that this is (t)). (c) Find the general solution to the original non-homogeneous system. ((1) Find the Jordan Form for the matrix A