Differential Equations : Solution Curves

[mehtamonica]

I have to solve the differential equation (y')^2= 4y to verify the general solution curves and singular solution curves.
Determine the points (a,b) in the plane for which the initial value problem (y')^2= 4y, y(a)= b has
(a) no solution ,
(b) infinitely many solutions (that are defined for all values of x )
(c) on some neighborhood of the point x=a , only finitely many solutions.

general solution that i am getting is y (x) = (x-c)^2 and singular solution is y(x)=0.

I am able to get part (a), as if b < 0, the problem has no solution.

Please help me figure out (b) and (c) .

 

[LCKurtz]

Think about a function g(x) defined piecewise with g(x) = 0 for x < c and g(x) = (x-c)² if x ≥ c.

 

[HallsofIvy]

For (c) consider the situation when b= 0.