check for the following if they have a single solution on (-1,1)
A)
$\displaystyle y'=\sqrt{|y|}$
y(0)=0
B)
$\displaystyle y'=|y|^{3}$
y(0)=0
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?
for A :
$\displaystyle dy/dx=\sqrt{|y|}$
$\displaystyle \int \frac{dy}{\sqrt{|y|}}=\int dx$
i dont know how to deal with the absolute value
ill just egnore it
$\displaystyle 2y^{\frac{1}{2}}=x$
$\displaystyle 4y=x^2 +c$
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
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.