I came across the following Ricatti-type ODE in my reading $$ \begin{aligned} \partial_t \psi(t,x) &= \Psi(\psi(t,x)),\\ \psi(0,x)&=x,\\ \Psi(x)&\triangleq \partial_t\psi(t,x)|_{t=0^+}. \end{aligned} $$

I know that if we augment the system by adding the requirement that $$

\begin{aligned} \partial_t \phi(t,x) &= \Phi(\psi(t,x)),\\ \phi(0,x)&=0,\\ \Phi(x)&\triangleq \partial_t\phi(t,x)|_{t=0^+}, \end{aligned} $$ then it becomes a generalized Riccati system in the sense of this PhD theis and can be shown to have a solution.

However, I'm wondering, without this additional condition it should always be solvable; is this indeed true and if so are the general solutions known?


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