# Solving an infinite non autonomous system of differential equations.

For all $\lambda\in\mathbb{R}$, let $J(\lambda)$ be the infinite matrix where $(J(\lambda))_{nn}=\lambda$, $(J(\lambda))_{n,n+1}=1$ for all $n\in\mathbb{N}$, and all other entries are $0$. This matrix can be seen as the infinite version of a Jordan block.

Let $\lambda(t)$ be a bounded, continuous, real-valued function. Can a unique solution be found for the following system: $$\frac{d\textbf{v}}{dt} = J(\lambda(t))\textbf{v},\hspace{0.5in} \textbf{v}(t_0)=\textbf{v}_0,$$ where $\textbf{v}$ is an unknown sequence of differentiable functions and $t_0$ and $\textbf{v}_0$ are given? If not, are there any more restrictions we can put on $\lambda(t)$ so that a unique solution does exist?

I'm not very well-versed in infinite systems of differential equations. This seems to be an elementary non autonomous system, but I'm not sure how to solve it, or if it's even possible. Any solutions, hints, or suggestions for references are appreciated.