Assume the ODE
$$ \frac{dy}{dt} = f(y) $$
Suppose $f$ is non linear. Is it possible that a change of parameterisation and a change of variables provide a linear right hand side ?
Let $$ y = \phi(z) $$ then the resulting ODE is $$ \dot z = \left[\frac{\partial \phi}{\partial z}(z)\right]^{-1} f(\phi(z)) $$
Now what if we take $\phi\in C^1$ diffeomorphism satisfying $$ \frac{\partial \phi}{\partial z} = f $$
that is $\phi$ is a primitive of $f$ ...