Possible solutions to the initial value problem

Question

Possible solutions to the initial value problem
Y'=y^1/2
y(0)=0

Answer 1

Separation of variable:

$ \int \frac{1}{\sqrt{y}} dy = \int 1 dx+c$. Hence

$2\sqrt{y}=x+c$. From $y(0)=0$ we get $c=0$. Therefore $y(x)=\frac{1}{4}x^2$.

But there are a lot of other solutions ! For example: $y(x)=0$ is asolution or , with $a>0$:

$y_a(x)=\frac{1}{4}(x-a)^2$ for $x \ge a$ and $y_a(x)=0 $ for $0 \le x \le a$ is a further solution.