I'm taking a first course in DE. Is this problem in the form $\displaystyle \frac{dy}{dt}+ay=g(t)$? How do I solve it?
$\displaystyle y\prime + 3y = t + e^{-2t}$
$\displaystyle y' + ay = g(t)$ .........................multiply through by $\displaystyle e^{at}$
$\displaystyle \Rightarrow e^{at}y' + ae^{at}y = e^{at}g(t)$ .........simplify the left hand side
$\displaystyle \Rightarrow (e^{at}y)' = e^{at}g(t)$ ..................integrate both sides
$\displaystyle \Rightarrow e^{at}y = \int e^{at}g(t)~dt$ .............solve for $\displaystyle y$
$\displaystyle \Rightarrow y = \frac {\int e^{at}g(t)~dt}{e^{at}}$