# Toronto Math Forum

## MAT244--2019F => MAT244--Test & Quizzes => Quiz-1 => Topic started by: hz12 on September 26, 2019, 11:50:11 PM

Title: TUT5103 quiz1
Post by: hz12 on September 26, 2019, 11:50:11 PM
\begin{aligned}
t y^{\prime}+2 y&=\sin (t), t>0\\
y^{\prime}+\frac{2}{t} y&=\frac{\sin (t)}{t}\\
y^{\prime}+p(t) y&=g(t)
\end{aligned}
$$\therefore P(t)=\frac{2}{t} \quad g(t)=\frac{\sin (t)}{t}$$
$$\mu(t)=e^{\int \frac{2}{t} d t}=e^{2 \ln (t)}=t^{2}$$
Muttiply $\mu(t)$ to standard equation
$$\begin{array}{c}{t^{2} y^{\prime}+2 t y=t \sin t} \\ {\mathrm{and}\left(t^{2} y\right)^{\prime}=t \sin t}\end{array}$$
Integrating  both  sides:
$$\int\left(t^{2} y\right)^{\prime}=\int t \sin t$$
Thus, $$t^{2} y=\int t \sin t$$
Integrating by parts:
\begin{aligned} u&=t, \quad d V=\sin t\\ d u&=d t \quad V=-\cos t \end{aligned}
\begin{aligned} \therefore \int t \sin t &=u v-\int v d u \\ &=-t \cos t-\int-\cos t d t \\ &=-t \cos t+\int \cos t d t \\ &=-t\cos t+\sin t+C \\ \end{aligned}
\begin{aligned}
\therefore t^{2} y&=-t \cos t+\sin t+C\\
y&=\displaystyle\frac{-t \cos t+\sin t+C}{t^{2}}
\end{aligned}

since given $t>0$, $y \rightarrow 0$ as $t \rightarrow \infty$

Title: Re: TUT5103 quiz1
Post by: Xinqiao Li on September 27, 2019, 10:45:42 AM
a very small typo: the final solution should be just y, not y^2.