Day Section Problem 1

[Victor Ivrii]

Find the general solution of
\begin{equation*}
y'''-y''+y'-y=e^{-t}\sin(t).
\end{equation*}

[Changyu Li]

$$
(r-1)(r^2+1) = 0 \\
r = 1, \pm i \\
y_h = c_1 e^t + c_2 e^{it} + c_3 e^{-it} \\
y_p = A e^{-t} \sin t + B e^{-t} \cos t \\
y_p' = e^{-t} \left(\left(A-B\right) \cos t - \left(A+B\right) \sin t \right) \\
y_p'' = -2 e^{-t} \left(A \cos t - B \sin t\right) \\
y_p''' = 2 e^{-t}\left(\left(A-B\right) \sin t + \left(A+B\right) \cos t \right) \\
A = 0, B = -\frac{1}{5} \\
y = c_1 e^t + c_2 e^{it} + c_3 e^{-it} -\frac{1}{5}e^{-t} \cos t
$$

[Victor Ivrii]

Please do not increase font size; also in this and another Quiz 3 problem provide solution in the real form as combining complex exponents and $\sin$. $\cos$ creates an eclectic mess