Transform the given differential equation or system into an equivalent system of first-order differential equations.

\(\displaystyle{x}^{{{\left({4}\right)}}}+{3}{x}{''}+{x}={e}^{{{t}}}{\sin{{2}}}{t}\) The fourth-order equation \(\displaystyle{x}^{{{\left({4}\right)}}}+{3}{x}{''}+{x}={e}^{{{t}}}{\sin{{2}}}{t}\)

is equivalent to system

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

Hense the substitutions \(\displaystyle{x}_{{1}}={x},{x}_{{2}}={x}'={x}'_{{1}},{x}_{{3}}={x}{''}={x}'_{{2}},{x}_{{4}}={x}^{{{\left({3}\right)}}}={x}'_{{3}}\)

yield the system

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

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

\(x'_3=x_4\)

\(\displaystyle{x}'_{{4}}={e}^{{{t}}}{\sin{{2}}}{t}-{x}_{{1}}-{3}{x}_{{3}}\) is a system of first-order equation.