Consider the linear system
where the -vector and the matrix are given; has constant real entries, and has distinct eigenvalues and linearly independent eigenvectors . Find the complementary function. Given a particular integral , write down the general solution. In the case show that the complementary function is purely oscillatory, with no growth or decay, if and only if
Consider the same case with trace and and with
where are given real constants. Find a particular integral when
(i) and ;
(ii) but .
In the case
with , find the solution subject to the initial condition at .