\(\displaystyle{y}{''}+{3}{y}'+{4}{y}={2}{\cos{{2}}}{t}\)

\(\displaystyle{y}{''}=-{3}{y}'-{4}{y}+{2}{\cos{{2}}}{t}\)

We use the substitution

\(y'=v\)

\(\displaystyle\Rightarrow{y}{''}={v}'\)

We substitute y' and y'' in equation

\(\displaystyle{y}{''}=-{3}{y}'-{4}{y}+{2}{\cos{{2}}}{t}\)

we get

\(\displaystyle{v}'=-{3}{v}-{4}{y}+{2}{\cos{{2}}}{t}\)

Therefore, we get system of first-order equations

\(y'=v\)

\(v'=-3v-4y+2\cos 2t\)