I know the theory needed to solve nonhomogeneous linear differential equations looking like $$\dot{x}(t) = A(t)x + b(t)\tag{1}$$ in finite-dimensional spaces.
Let $A(t)$ be a linear form, i.e. mapping into $\mathbb{R}$, and $b$ be real-valued. If I have a monotonous smooth diffeomorphism $\varphi\in\operatorname{Diff}^\infty(\mathbb{R},[0,1])$, how does solving $(1)$ relate to solving the possibly non-linear differential equation $$\dot{x}(t) = \varphi\left(A(t)x + b(t)\right),\tag{2}$$ in terms of solvability, stability etc. Do the solutions of $(1)$ and $(2)$ relate at all?
Many thanks in advance.