Consider the system
(a) Show that is a triple eigenvalue of the coefficient matrix
and that there are only two linearly independent eigenvectors,
which we may take as
Find two linearly independent solutions and
of Eq. (i).
(b) To find a third solution assume that thow that and must satisfy
(c) Show that where
and are arbitrary constants, is the most general solution of Eq.
(iii). Show that in order to solve Eq. (iv) it is necessary that
(d) It is convenient to choose For this choice show that
where we have dropped the multiples of and that appear
in . Use the results given in Eqs. (v) to find a third linearly
independent solution of Eq. (i).
(e) Write down a fundamental matrix for the system (i).
(f) Form a matrix T with the cigenvector in the first column and
with the eigenvector and the generalized eigenvector from Eqs.
(v) in the other two columns. Find and form the product
. The matrix is
the Jordan form of