Question: I need to prove that the following ODE do not have solution in Real numbers domain:
$$
\begin{cases}
\big(y'\big)^2=y-\lvert x\rvert\\[1.2ex]
y(x=0)=0
\end{cases}
$$
I found the complex solution but have trouble in explaining why there is no real solution.

Answer: 
Take a collection of functions and see if you can demonstrate all the properties of a vector space using them.

A vector space requires:

1. An additive identity (written $0$ in ${\mathbb R}$). The function $f\equiv 0$ fulfills this need.
2. A scalar multiplicative identity (written $1$ in ${\mathbb R}$). $1$ works here since $1\cdot f = f$
3. Commutativity of addition: $f+g = g+f$
4. Associativity of addition: $f+(g+h) = (f+g)+h$
5. Associativity of scalar multiplication: $\alpha (\beta f) = (\alpha \beta)f$
6. Distributivity of scalars: $(\alpha + \beta)f = \alpha f + \beta f$
7. Distributivity of scalars over vector addition: $\alpha(f+g) = \alpha f + \alpha g$
8. An additive inverse: given $f$ there exists $g$ such that $f+g = 0$. Obviously $g=-f$ satisfies this.
The notion of a vector space is abstract and it can be applied to Functions (sometimes these spaces are called Function Spaces) and these spaces are the subject of Functional Analysis. 

It is important to get away from the geometric representation of a vector space - they are special cases and impossible to think about when we move into higher dimensional spaces. Instead, consider our definition of a vector space:

Let $V$ be a set which is closed under vector addition and multiplication of vectors by scalars, then we call V a vector space. i.e.

$$ \forall x, y \in V \quad \forall c,d \in R $$

$$ cx+dy \in V $$

When we are talking about function spaces we are talking about mappings from a set $X$ to a vector space (over a field but don't get too bugged down in this if you don't know what this means) $V$ (note that $X$ can also be a vector space, then we can consider the function space to be a linear mapping). 

A more simple way of thinking about them is considering them as a collection of functions that share characteristics of their range and whose codomains' are commonly the same.

Example time! 

The function space $C(\mathbb R^n)$ represents all functions who are continuous in $\mathbb R^n$. e.g. $f(x)=x \in C(\mathbb R^1 )$

A Hilbert Spaces are subspaces of $C(\mathbb R^n)$ and often referred to as generalisations of Euclidean space.