Problem ordinary differential equation Consider the ordinary differential cquation
$y^{\prime}=y(y-1)(y-2)$
Which of the following statements is true ?
(a) If $y(0)=0.5$ then $y$ is decreasing
(b) If $y(0)=1.2$ then $y$ is increasing
(c) If $y(0)=2.5$ then $y$ is unbounded
(d) If $y(0)<0$ then $y$ is bounded below
I solve using separation  of variable and get implicit relation as
$\frac{ y(y-1)}{(y-2)^2}= A e^ {2x} $
Now this becomes very difficult to apply condition and find the value of $A$  and check option
is there easy method  to solve such questions ?

There is no need to actually solve the differential equation—you can extrapolate the behavior of the solution by looking at the behavior of $y’$ for the given initial conditions.
For example, consider the first two statements regarding whether $y’$ is increasing or decreasing for given values of $y$. Simply insert the stated $y$ value into the expression for $y’$ and determine whether it’s positive or negative, and that tells you whether or not $y$ is increasing or decreasing.
For the latter two cases, the process is a little more subtle—you need to be able to note the regions of initial $y$’s where $y$ will continue to monotonically increase or decrease, and use that information to determine what behavior is true or false.
