6. Consider the ODE
(*)
x" - 6x' + 5x = 0
(a) Find the general solution to (*) by writing down the characteristic polynomial, finding its roots, and using
them in the appropriate formula.
(b) Find the solution of the initial value problem consisting of the ODE (*) and the initial conditions x(0) = 6,
x'(0) = 14.
(c) Find a matrix A such that (*) is equivalent to the first-order system \(\dot{X} = AX\), where \(X = \begin{bmatrix} x \ x' \end{bmatrix}\).
(d) Find the eigenvalues and eigenvectors of A. What is the relationship between the characteristic equation
of (*) and the equality 0 = det(A - \(\lambda I\))?
(e) Use your answer to (the first part of) (d) to write down the general solution to the ODE system \(\dot{X} = AX\).
(f) How do the initial conditions from (b) translate into initial conditions for X? Use your answer, together
with your answer to parts (c) and (d), to solve the initial value problem again. Show your work.
