Given the differential equation y$^(\text{'})=$y$(3-$y$),$

$($a$)$ Verify that y$=(3)/(3$Ce$^(-3$t$)+1),$ where C is a constant, is a one$-$parameter family of solutions.

$($b$)$ Find a solution from the family in part $($a$)$ that satisfies y$(0)=2.$ Determine the largest interval of definition for

that solution.

$($c$)$ Find a solution from the family in part $($a$)$ that satisfies y$(0)=4.$ Determine the largest interval of definition for

that solution.

$($d$)$ Find a solution that satisfies y$(0)=3.$ Hint: See Worksheet $1,$ Problem $5($b$).$