Differential equation for auditory system

Question

Let $x$ and $y$ be input and output signals, respectively. A simple low-pass filter satisfies

$$ y' = k (x - y) $$

where $k > 0$. A simple high-pass filter satisfies

$$ y = k (x - y)' $$

The gammachirp filter is a widely used model of the auditory system. Its impulse response is

$$ h(t) = \Re(\exp(z_1 + z_2 t + z_3 \ln t)) $$

where $z_1, z_2, z_3 \in \mathbb{C}$. What differential equation does it satisfy? In general, what differential equation do auditory filters satisfy, approximately?

Answer 1

$$t\frac{d}{dt}h(t)=(z_{2}t+z_{3})h(t)$$