Starting system is

\(\displaystyle{x}{'''}+{t}{x}{''}+{2}{t}^{{{3}}}{x}'-{5}{t}^{{{4}}}={0}\)

We continue as in example from the book

\(\displaystyle{x}'_{{3}}={x}{'''}\)

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

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

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

\(\displaystyle{x}'_{{2}}={x}_{{3}}\)

\(\displaystyle{x}'_{{3}}=-{2}{t}^{{{3}}}{x}_{{2}}-{t}{x}_{{3}}+{5}{t}^{{{4}}}\)

\(\displaystyle{x}{'''}+{t}{x}{''}+{2}{t}^{{{3}}}{x}'-{5}{t}^{{{4}}}={0}\)

We continue as in example from the book

\(\displaystyle{x}'_{{3}}={x}{'''}\)

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

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

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

\(\displaystyle{x}'_{{2}}={x}_{{3}}\)

\(\displaystyle{x}'_{{3}}=-{2}{t}^{{{3}}}{x}_{{2}}-{t}{x}_{{3}}+{5}{t}^{{{4}}}\)