<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand\ds{\displaystyle}
\begin{document}

\parindent=0pt

QUESTION

Find a particular integral of the differential equation
$\ds\frac{d^2x}{dt^2}+4x=e^{-t}$.

\bigskip


ANSWER


$\ds\frac{d^2x}{dt^2}+4x=e^{-t}$

To find a particular integral try $\ds x=Ce^{-t},\
\frac{dx}{dt}=-Ce^{-t},\ \frac{d^2x}{dt^2}=Ce^{-t}$\\ Substituting
this into the ODE gives:

 $Ce^{-t}+4(Ce^{-t})=5Ce^{-t}=e^{-t}$
therefore $\ds C=\frac{1}{5}$

Hence a particular integral is $\ds x=\frac{1}{5}e^{-t}$.




\end{document}
</pre></body></html>