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3
By default, NDSolve uses a "shooting" method to satisfy the boundary conditions: it picks an initial condition and then evolves the equation to see what boundary values are produced by the initial conditions. It then rejiggers these initial conditions to produce the boundary values specified by the boundary conditions. Here is the Mathematica documentation ...
2
When you try to find the general solution with initial condition y[x0]==y0 you see what happens
sol = y[x] /.
First@DSolve[{y'[x] - x^2 (y[x] + 1) (y[x] - 2)^2 == 0, y[x0] == y0}, y[x], x]
As you see when y0==2 is directly inserted into the solution the denominator becomes zero. At this point you could use Limit to investigate further.
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