The function family
y=(C1+C2x)e3x
contains two arbitrary constants C1 and C2.
(1) Part 1: Determine y′, y′′, and y′′′ for the general member of the family.
(2) Part 2: Eliminate the constants C1 and C2 to obtain a differential equation whose solution set is exactly the given family. Present your result in the form ay′′′+by′′+cy′+dy=0, where a, b, c, and d are real numbers.
(3) Part 3: State the order of the differential equation found in part 2 and justify why no differential equation of lower order can generate the entire family.