Differentiability of flow of non-autonomous ODE I am looking for any reference on the regularity of flows of non-autonomous ODEs. Any suggestions on books or articles are very much welcome. I have had little luck finding any usable results.
Let us assume that
$$
f: [0,1]\times \mathbb{R}^N \to \mathbb{R}^N
$$
is continuous, satisfies the Lipschitz condition (i.e., it is Lipschitz continuous in the second argument), and is continuously differentiable in the second argument. Furthermore, assume that its derivative in the second argument, $\frac{\partial f}{\partial x}$, also satisfies the Lipschitz condition.
Consider the flow $\varphi : [0, 1] \times \mathbb{R}^N \to \mathbb{R}^N$ of the ODE
$$
\dot x = f(t, x)
$$
Meaning that $\varphi$ is defined via:
\begin{equation}
\begin{cases}
\frac{d}{dt}\varphi(t, x_0) = f(t, \varphi(t, x_0)) \\
\varphi(0, x_0) = x_0
\end{cases}
\end{equation}
For fixed $t\in[0,1]$, I expect that $\varphi(t, \cdot):\mathbb{R}^N \to \mathbb{R}^N$ is differentiable. In fact, I would expect the derivative $\frac{\partial \varphi}{\partial x}(\cdot, x_0)$ to satisfy the usual variational equation
\begin{equation}
\begin{cases}
\frac{d}{dt}X(t, x_0) = \frac{\partial f}{\partial x}(t, \varphi(t, x_0))X(t, x_0) \\
X(0, x_0) = I
\end{cases}
\end{equation}
I have been searching through dozens of books for any result like this. Most books, e.g., "Differential Equations, Dynamical Systems & An Introduction to Chaos", only consider autonomous ODEs with a right-hand side which is $C^1$. Oftentimes, non-autonomous systems are dismissed at the very beginning with the note that it is possible to reduce to the autonomous setting. In my case, this will not be possible since the autonomized right-hand side will not be in $C^1$.
I would be grateful if somebody could point me towards helpful literature. Thanks!

Consider the following system:
$$\begin{cases}
\dot{\tau} = \omega\\
\dot{\omega} = 0
\end{cases}$$
with initial conditions $\tau(0) = 0$ and $\omega(0) = 1.$
The solution is $\tau(t) = t$ and $\omega(t) = 1.$
Now, consider the following system:
$$\begin{cases}
\dot{x}_1 = f_1(t, x_1, \ldots, x_n)\\
\ldots \\
\dot{x}_n = f_n(t, x_1, \ldots, x_n)\\
\end{cases}.$$
You can add the equations of $\tau$ and $\omega$, and replace the explicit $t$ with $\tau$:
$$\begin{cases}
\dot{x}_1 = f_1(\tau, x_1, \ldots, x_n)\\
\ldots \\
\dot{x}_n = f_n(\tau, x_1, \ldots, x_n)\\
\dot{\tau} = \omega\\
\dot{\omega} = 0
\end{cases}.$$
In this way, you are able to  transform a non-autonomous system into an autonomous one.
