$$\begin{align}\frac{dy}{dx} &= y\\[1em] \frac{dy}{y} &= dx \,\,\,\,\qquad\qquad\Bigl \{y \neq 0\\[1.5em] \int \frac{dy}{y} &= \int dx\\[1.5em] \ln{|y|} &= x + c\\[1em] |y| &= e^{x+c} = ae^x \quad \Bigl\{a>0\\y &= Ae^x \qquad\qquad \Bigl \{A \in \mathbb{R}\end{align}$$
Note: $A$ can be both positive and negative, since $|y| = ae^x$ and $a>0$. So, when we remove the absolute value function from $|y|$ to get $A$, we get both positive and negative values. Also, $A$ can be zero even though $a>0$. This is because when we divided by $y$ in our 2nd step, we lost the $y=0$ solution.
Now, given our starting condition $y(0) = 0$, we can calculate the value of $A$.
So, putting $x=0$ and $y(0)=0$:
\begin{align} 0 &= Ae^0\\[1em]0 &=A\cdot 1\\[1em]\implies A &= 0\end{align}
Thus, the solution to the given ODE is zero.
Hence, option D is the correct answer.
