An ODE involving bump functions Consider the following initial value problem
$$ 
\begin{cases}
\frac{d}{dt} y(t) = \rho(y(t))\\
y(0) = 0
\end{cases}
$$
where $\rho(x)$ is a bump function supported near $0$ on $\mathbb{R}^1$.
That is, $\rho(x)$ is a $C^\infty$ function on $\mathbb{R}^1$ 
such that $\rho(x) \geq 0$ and
$\operatorname{supp} (\rho(x) ) \subseteq (a,b)$ for some bounded 
open interval $(a,b)$ containing $0$. 
By abstract reasoning, a solution $y(t)$ exists
and the image of $y(t)$ is bounded.
I'm wondering if it's actually possible to solve $y(t)$ in an exact form. 
Say if $\rho(x)$ is instead an strictly increasing function larger than $0$. 
Then the naive dividing-$\rho(y(t))$-on-both-side method can lead to
an somewhat exact expression of $y(t)$. 
However, as in any first ODE course, 
this is not what we are supposed to do. So I wonder if there is a similar
method to the one for ODE of the form:
$$ 
\begin{cases}
\frac{d}{dt} y(t) = f(t)y(t)\\
y(0) = 0
\end{cases}
$$
 A: See separation of variables:
https://en.wikipedia.org/wiki/Separation_of_variables
The "same" method works, indeed divide by $f(y)$ then the solutions satisfy $g(y(t)) = t + c$ ever $g$ is a primitive of $1/f$. You need to analyze this carefully for any particular example.
In a bit more detail: Consider$$ 
\begin{cases}
\frac{d}{dt} y(t) = \rho(y(t))\\
y(0) = 0
\end{cases}
$$I understand that, in the case in which $\rho$ is positive, we can get an "explicit" solution by dividing by $\rho$, and introducing a new function given by the integral of $1/\rho$. In the case, you described, in which $\rho$ is a bump function, then, I agree, the solution $y(t)$ is bounded. But, I claim, that solution also has the following further property: the image of $y(t)$ (as $t$ varies) lies entirely within the interior of the support of $\rho$. Hence, we can again divide by $\rho$, and again use the integral of $1/\rho$, to get an "explicit" solution.

Exercise. Proof or counterexample: For any continuous function $\rho$, $$\int(\text{support}\, \rho) = \text{set of points at which }\rho\text{ is positive}.$$

