Starting system is

\(\displaystyle{x}{''}-{3}{x}'+{2}{x}={t}^{{{2}}}\)

We continue as in example from the book

\(\displaystyle{x}'_{{2}}={x}{''}\)

\(\displaystyle{x}'_{{1}}={x}'\)

\(\displaystyle{x}'_{{1}}={x}_{{2}}\)

\(\displaystyle{x}'_{{2}}=-{2}{x}_{{1}}+{3}{x}_{{2}}+{t}^{{{2}}}\)