The ODE
$\frac{dy}{dt} = r(g(t) - y)$
tends to "stabilize" $y$ when $r$ is a positive constant, and makes $y$ move toward the current value of $g(t)$.
I'm trying to construct a reasonably easy-to-solve ODE that does the exact same thing, but makes $y$ move toward $g(t)$ far faster when $y$ is below $g(t)$ than above it. If such an ODE exists and one made $\frac{dy}{dt}$ high enough when $y$ is below $g(t)$, this would make $g(t)$ a sort of floor for $y$.
To illustrate, if $g(t)$ were constant some potential paths of $y$ might look like this:
The black line is $g(t)$. The blue paths and orange paths are both sets of solutions to the desired ODE, but as you can see, the orange paths converge much faster because their initial condition $y(t_{0})$ was lower than $g(t_{0})$.
Are there are first-order ODEs that display this "asymmetric stabilization" behavior, that aren't too difficult to work with (i.e. it's possible to find a closed-form solution)? I'm searching for something that works with an arbitrary piecewise-continuous function $g$, rather than the constant function I used in the example.