ok, so how does one attempt this using separation of variables?:
y' = the square root of ty
so far, what i have done is set y' equal to t^(1/2) * y^(1/2) since the square root of two variables is just the square root of each individual variable.
i'm not really sure what to do after that so i would be grateful for any help!
There is actually a stranger situation, that might occur. I addressed it on this site some time ago, when y=0 at some point on the interval. My belief is that y=0 either everywhere on the interval or y not = 0. Meaning y=0 at some points yes and some points no, cannot happen. I never been able to formally prove it, but my idea is that when a curve y not = 0 for some point, then around that neighborhood of points we have a solution (which is of the form you mentioned it) now if it is zero at some other point and you connect the non-trivial solution (of the form you mentioned) with a curve, we get a "rough" point, you know a sharp turn, and therefore it is non-differenciable at that point. But that cannot happen because by definition a solution to some differencial equation must be differenciable. This tells us a very useful fact (assuming it is true), only consider the case y=0 before you divide.