Moderate# Nonhomogeneous Linear System: Undetermined Coefficients

DIFFEQ-@FHWS3

Consider $x'=Ax+\left(\begin{array}{c}t^2\\\ 2t\end{array}\right)$, where $ A=\left(\begin{array}{cc} 3&-4\\\ 1&-1\end{array}\right).$

The general solution can be written as $x=x_h+x_p$ where $x_h$ is a general solution of $x'=Ax$ and $x_p$ is a particular solution. One can find a particular solution by variation of parameters or by undermined coefficients. In the latter method, we simply guess a form $x_p= a t^2+b t+c$ where $a,b,c$ are three vectors.

The three vectors are given by