So I am assigned the task of finding the particular solution for the given problem: $$ y' = 1/t \begin{bmatrix} 1 & t\\ -t & 1\\ \end{bmatrix} + t \begin{bmatrix} cost\\ sint\\ \end{bmatrix} $$
You are told that a fundamental matrix for the complementary system is: $$ Y = t\begin{bmatrix} cost & sint\\ -sint & cost\\ \end{bmatrix} $$
I know that I can arrive at my particular solution given this and the f(t) provided in the problem statement through the following process: $$ u' = Y^{-1}f $$
The particular solution is then given by: $$ X_p = Yu $$
I have arrived at a particular solution of: $$ X_p = \begin{bmatrix} tsint\\ -tcost\\ \end{bmatrix} $$
However, the book states that the particular solution should be: $$ X_p = \begin{bmatrix} tsint\\ 0\\ \end{bmatrix} $$
Could someone verify that this is the correct particular solution? I have gone through my process twice and arrived at the same answer. Perhaps someone could shed some light on where I may be going wrong?
Thanks!