\documentclass[a4paper,12pt]{article}
\begin{document}


{\bf Question}

Find the complementary function for the  equation:
$$\frac{d^2x}{dt^2} + 3 \frac{dx}{dt} + 2x = 1 + 2t + t^2$$


\vspace{.25in} {\bf Answer}

The complementary function for $\displaystyle \frac{d^2x}{dt^2} +
3 \frac{dx}{dt} + 2x = 1 + 2t + t^2$

Is solution of $\displaystyle \frac{d^2x}{dt^2} + 3 \frac{dx}{dt}
+ 2x = 0$

Auxiliary equation is $\displaystyle m^2 + 3m + 2 = (m + 2)(m + 1)
= 0 \Rightarrow m = -2, -1$

Hence $$x_c = Ae^{-2t} + Be^{-t}$$

with A, B arbitrary constants.


\end{document}