Suppose [tex]y_p=a_0+a_1t[/tex] is a solution to the ODE. Then [tex]{y_p}'=a_1[/tex] and [tex]{y_p}''=0[/tex], and substituting these into the ODE gives
[tex]a_1-4(a_0+a_1t)=7t+5\implies\begin{cases}-4a_1=7\\-4a_0+a_1=5\end{cases}\implies a_0=-\dfrac{27}{16},a_1=-\dfrac74[/tex]
Then the particular solution to the ODE is
[tex]y_p=-\dfrac{27}{16}-\dfrac74t[/tex]
