Htykuut

In most analysis courses one sees that differential equations of order $n$ are basically a subset of higher dimensional differential equations of order $1$, for example the equation:

$$f^{(n)}(t)=Fleft(f(t),f'(t),...,f^{(n-1)}(t),tright)$$

Is the same as:
$$frac{d}{dt},begin{pmatrix}g_0(t)\g_1(t)\vdots\g_{n-1(t)}end{pmatrix}=begin{pmatrix}g_1(t)\vdots\g_{n-1}(t)\Fleft(g_0(t),g_1(t),...,g_{n-1}(t),tright)end{pmatrix}$$

This is especially useful, as it allows one to write down explicitly the solutions to linear differential equations of finite order, if we have:
$$f^{(n)}(t)=sum_{k=0}^{n-1}a_k, f^{(k)}(t)$$
Then the corresponding $n$-dimensional equation is of the form:
$$frac{d}{dt} g(t) = Acdot g(t)$$
for some matrix $A$ and the solution is $g(t)=exp(A,t)g(0)$. It is possible to generalise to time dependent coefficients $a_k$.

Is there a way to implement this trick for differential equations that are essentially of infinite order? For example the equation
$$f=sum_{k=1}^infty f^{(k)}(t),$$
of which the solution space is $f={Cexp(frac t2)mid Cinmathbb R$ (or $mathbb C$)$}$. More generally I would like to put something of the form
$$sum_{k=0}^infty a_k, f^{(k)}(t)=0$$
(where $a_k$ are as regular as needed (but with infinite non-zero terms)) into the form
$$frac{d}{dt} u = A(u)$$
Where $u$ is a map $C^infty(mathbb R,X)$ with $X$ a Banach space and $Ain mathcal L(X)$.