By recursively integrating according to $x{n+1}!\left(t\right):=\int{t_0}^tf!\left(x_n!\left(s\right),s\right)\,\mathrm ds+C$ from $x_0!\left(t_0\right):=C$, we can get the solution of the ODE $x’!\left(t\right)=f!\left(x!\left(t\right),t\right)$ with initial conditions $x!\left(t_0\right)=C$ as the limit of the sequence of functions.
https://ulysseszh.github.io/math/2022/11/15/ode-recursive.html
By recursively integrating according to $x{n+1}!\left(t\right):=\int{t_0}^tf!\left(x_n!\left(s\right),s\right)\,\mathrm ds+C$ from $x_0!\left(t_0\right):=C$, we can get the solution of the ODE $x’!\left(t\right)=f!\left(x!\left(t\right),t\right)$ with initial conditions $x!\left(t_0\right)=C$ as the limit of the sequence of functions.