Overlooked solutions in a simple first order DE

Question

When we have simple equations like $\frac{dy}{dx}=\frac{3(y-1)(y-5)}{x}$ we tend to separate the variable by dividing over by $(y-1)(y-5)$ and then do partial fractions and so on(by we I mean DE calculators as well). But aren't $y=1$ and $y=5$ also solutions since they satisfy the DE?

Answer 1

You are right, they are solutions too. There are two points I would keep in mind:

1) If you separate variables, would you get the solutions y=1 and y=5 as members of the family of solutions?

2) The initial conditions play a key role in whether there exists or not a solution, so it is possible that variables separation does not allow you to get a solution with initial condition y=1 or y=5.