Toronto Math Forum
MAT2442013S => MAT244 MathTests => Quiz 3 => Topic started by: Victor Ivrii on February 27, 2013, 07:46:14 PM

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

$$
(r1)(r^2+1) = 0 \\
r = 1, \pm i \\
y_h = c_1 e^t + c_2 e^{it} + c_3 e^{it} \\
y_p = A e^{t} \sin t + B e^{t} \cos t \\
y_p' = e^{t} \left(\left(AB\right) \cos t  \left(A+B\right) \sin t \right) \\
y_p'' = 2 e^{t} \left(A \cos t  B \sin t\right) \\
y_p''' = 2 e^{t}\left(\left(AB\right) \sin t + \left(A+B\right) \cos t \right) \\
A = 0, B = \frac{1}{5} \\
y = c_1 e^t + c_2 e^{it} + c_3 e^{it} \frac{1}{5}e^{t} \cos t
$$

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