\(\displaystyle{x}{''}+{3}{x}'+{7}{x}={t}^{{{2}}}\)

is equivalent to equation

\(\displaystyle{f{{\left({t},{x},{x}'\right)}}}={t}^{{{2}}}-{7}{x}-{3}{x}'\)

Hence the substitutions \(\displaystyle{x}_{{{1}}}={x},{x}_{{{2}}}={x}'={x}_{{{1}}}\) yield the system

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

\(\displaystyle{x}'_{{{2}}}={t}^{{{2}}}_{\left\lbrace{1}\right\rbrace}-{7}{x}_{{{1}}}-{3}{x}_{{{2}}}\)