There's a lot going on with this one. Three things:
(1) You can solve it directly by separation of variables as you mentioned. I get:
(2) You can solve it numerically with Euler or other means. Not too hard to set up Euler right? Or if you've done a few Euler's, then you can set it up in Mathematica say for the interval (0,5):
Code:
sol = NDSolve[{Derivative[1][y][t] == 5 - 3*y[t]^(1/2),
y[0] == 2}, y, {t, 0, 5}]
p1 = Plot[Evaluate[y[t] /. sol], {t, 0, 5}]
Yeah, I know. It's almost a crime to do it that way.
(3) You can extract an explicit expression for y in terms of x in (1) using the Lambert function. Suppose:
Then to solve for y in terms of x, it's:
So if I have:
Can you show:
However if you're like me you're skeptical about all this. So I plotted the NDSolve solution and the Lambert W solution and superimposed them. The plot is below. Works for me.