#2:
I get the following solutions for the respective DE's:
Perhaps that is something for you to start with.
for your first question, since is not an eigenvalue of A, the matrix is invertible. let
and then and hence: which is
what you were asked to prove. now, the example in your first question should be very easy for you to do!
for the second question, call the first differential equation (1), the second (2) and the third (3). first differentiate
both sides of (1) with respect to to get: call this (4). now plug from (2) into (4) to get
which is a simple linear equation and gives you the general solution for now that we
have we can easily find and because by (1): and by (3):