Toronto Math Forum

MAT244--2019F => MAT244--Test & Quizzes => Quiz-1 => Topic started by: Huyi Xiong on September 27, 2019, 02:50:13 PM

Title: TUT0801 QUIZ1
Post by: Huyi Xiong on September 27, 2019, 02:50:13 PM
\begin{align}
y'+\frac{1}{t}y &= 3cos(2t)\\
p(t) &= \frac{1}{t} \\
m &= e^{-\int {p(t)} dt} \\
&= e^{-\int {\frac{1}{t}} dt} \\
&= e^{-lnt} \\
&= \frac{1}{t} \\
n &= \int {\frac{3cos(2t)}{\frac{1}{t}}} dt \\
&=  \int {3tcos(2t)} dt \\
\textbf{Integrating by parts:} \\
u = t, du = dt, v = \frac{1}{2}sin(2t), dv = cos(2t)dt \\
n &= 3[\frac{1}{2}tsin(2t)- \int {\frac{1}{2}sin(2t)} dt] + c \\
&= 3[\frac{1}{2}tsin(2t)+ \frac{1}{4}cos(2t)] + c \\
y &= mn \\
&= \frac{1}{t}[3[\frac{1}{2}tsin(2t)+ \frac{1}{4}cos(2t)] + c] \\
&= \frac{3}{2}sin(2t) + \frac{3cos(2t)}{4t} + \frac{c}{t}
\end{align}