1. ## single solution

check for the following if they have a single solution on (-1,1)
A)
$y'=\sqrt{|y|}$
y(0)=0
B)
$y'=|y|^{3}$
y(0)=0

2. In order to find a singular solution, it must be referenced to a family of solutions that you've already found. That is, a singular solution is a solution that can't be expressed as a member of a family of solutions. What is the family of solutions you've already found?

3. for A :
$dy/dx=\sqrt{|y|}$
$\int \frac{dy}{\sqrt{|y|}}=\int dx$
i dont know how to deal with the absolute value
ill just egnore it
$2y^{\frac{1}{2}}=x$
$4y=x^2 +c$

by single solutioni didnt mean singluar solution which for 1/x x=0 is a singular point of solution

i ment like in lenear algebra where we get a single solution from a row redused matrice

4. My mistake. I see what you mean now.

I think you're off by a factor of 2 in your solution. You should always plug your solution back into the DE to make sure it solves the DE. It's easy and it prevents mistakes. I also think that y = 0 is a perfectly good solution for both (A) and (B). Hence, I think both of them have multiple solutions. Indeed, the conditions of the standard existence theorem are not satisfied, so you're not guaranteed a unique solution.

5. you are correct i fixed the original post
so y=0 is the single solution
?

6. I would say that y = 0 is a singular solution. You cannot get the solution y = 0 from any choice of C in the family of solutions $4y=x^2 +c.$

There is more than one solution to the IVP's for both (A) and (B).