# Toronto Math Forum

## MAT244--2019F => MAT244--Test & Quizzes => Quiz-1 => Topic started by: Zuwei Zhao on September 27, 2019, 02:00:13 PM

Title: MAT24f4 TUT5103 Quiz1
Post by: Zuwei Zhao on September 27, 2019, 02:00:13 PM
Find the general solution of the given differential equation in variation of parameter.
$$t y^{\prime}+2 y=\sin (t), t>0$$

$$y^{\prime}+\frac{2}{t} y=\frac{\sin (t)}{t}$$
$$\therefore P(t)=\frac{2}{t}$$
$$\therefore \mu(t)=e^{\int \frac{2}{t} d t}=e^{2 \ln t}=t^{2}$$
$$\text{multiply} \quad\mu(t)\quad \text{on both sides}$$
$$\begin{array}{l}{\therefore t^{2} y^{\prime}+2 t y=\sin (t) \cdot t} \\ {\because t^{2} y^{\prime}+2 t y=\left(t^{2} y\right)^{\prime}} \\ {\therefore\left(t^{2} y\right)^{\prime}=\sin (t) \cdot t}\\{\therefore \int\left(t^{2} y\right)^{\prime}=\int \sin (t) \cdot t} \\ {\text { let } u=t \quad d v=\sin (t)} \\ {\therefore d u=d t \quad v=-\cos (t)} \\ {\therefore t^{2} y=t \cdot(-\cos (t))-\int-\cos (t) d t} \\ {\quad y=\frac{-t \cdot \cos (t)+\sin (x)+c}{t^{2}}}\end{array}$$