$(10$ points$)$ The evolution of a certain population with a constant migration rate M and time$-$dependent reproductive rate K$($t$)=$ a $+$ b sin$($ct$)($where a$,$ b$,$ c are non$-$zero constants$)$ is described by the initial value problem dP $/$dt $=($a $+$ bsin$($ct$))$P$($t$)+$ M$,$ P $(0)=1000.(2)($a$)$ Determine the integrating factor for $(2).($b$)$ Use the integrating factor to solve the initial value problem $(2)$ and provide the solution P $($t$)$ in an integral form. $($c$)$ Determine P $2$c $.5$