Initial value problem of $y'(x)=2\cdot\sqrt{|y(x)|}$

Question

I have a problem to solve this initial value problem of $y'(x)=2\cdot\sqrt{|y(x)|}$ with $y(0)=0$.

How to solve initial value problems in general is clear to me (integrate and determine the constant). However, I don't know how to deal with the $y(x)$. I think this is a special case of initial value problems with an infinite number of solutions? How can I solve this problem?

(I cannot integrate $y(x)$ in the form in which the initial value problem is now presented, and how do I then determine the infinite solutions?).

Answer 1

I will assume $x\in(-\infty,\infty)$ and $y\in[0,\infty)$.

$$\frac{dy}{dx}=2\sqrt{y}$$

$$\Rightarrow\sqrt{y}=x$$

But we have $x\in(-\infty,\infty)$

so we must have

$$y=\begin{cases} 0 & x\leq x_0 \\ (x-x_0)^2 & x\geq x_0 \end{cases}$$

where $x_0$ is any real number.

So you do have infinite solutions here.