$V ( t ) = t \cos t , t \geq 0 , y ( 0 ) = 3$

a) $t \cos t > 0,0 \leq t \leq 5$

$t \cos t = 0$

$t = 0$

$$\begin{gathered}
\cos t = 0 \quad t \quad \cdots \quad t \\
t = \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \quad a _ { - } \frac { \pi } { 2 } \quad \frac { \pi } { 2 } \quad E
\end{gathered}$$

$( 0 , \pi / 2 ) , ( 3 \pi / 2,5 )$

b) $$\begin{aligned}
\Lambda ( t ) & = V ( t ) \\
\Lambda ( t ) & = \cos t + t - \sin t ) \\
& = \cos t - t \sin t
\end{aligned}$$

() $Y ( t ) = \int Y ( t ) d t$

$$\begin{aligned}
& y ( t ) = \int t \cos t d t \text { let } \quad \frac { d } { d } = t \quad \frac { d v } { d } \\
& = \cos t \\
& = t \sin t - \int \sin t d t \quad \frac { d t } { d t } \quad v \\
&
\end{aligned}$$

$Y ( t ) = + \sin t - ( - \cos t ) + c$

$= t \sin t + \cos t + c$

$y | 0 | = 3$

$j = \cos 0 + c$

$c = 2$

$Y ( t ) = t \sin t + \cos t + 2$

d) $V ( t ) = t \cos t$

$t \cos t = 0$

$t \neq 0 \quad \cos t = 0$

$$t = \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } , \ldots$$

$Y \left( \frac { \pi } { 2 } \right) = \frac { \pi } { 2 } \sin \frac { \pi } { 2 } + \cos \frac { \pi } { 2 } + 2 = \frac { \pi } { 2 } + 0 + 2 = \frac { \pi } { 2 } + 2$