The solution curves for dy/dx = 2√ y will be solution curves for (dy/dx)2 = 4y. However, so will the solution curves for dy/dx = −2√y. So, no, they do not have the same solution curves. The function 2√y is continuous for y ≥ 0. Its partial derivative with respect to y, 1/√y , is continuous for y > 0.
The differential equation dy/dx = 2√y is separable, and we can solve it:
Solving this for y we get
y = (x + C)2
(a) - There will be no solution if b < 0.
(b) - There will be (locally) a unique solution for b > 0.
(c) - If b = 0 there will be infinitely many solutions.