# Can (linear) differential equations of infinite order be recast into equations of first order?

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:

$$f^{(n)}(t)=F\left(f(t),f'(t),...,f^{(n-1)}(t),t\right)$$

is the same as: $$\frac{d}{dt}\,\begin{pmatrix}g_0(t)\\g_1(t)\\\vdots\\g_{n-1(t)}\end{pmatrix}=\begin{pmatrix}g_1(t)\\\vdots\\g_{n-1}(t)\\F\left(g_0(t),g_1(t),...,g_{n-1}(t),t\right)\end{pmatrix}.$$

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: $$\frac{d}{dt} g(t) = A\cdot g(t)$$ 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 $$f=\{C\exp(\frac t2)\mid C\in\mathbb R$$ (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 $$A\in \mathcal L(X)$$.

• I'm still hoping someone to answer this question :)
– Our
Commented Feb 13, 2018 at 10:17

No, in general that is not possible. For instance you can write a delay-differential equation as such an "infinite order" equation using the Taylor expansion, $$y'(t)=f(t,y(t),y(t-r)) =f\left(t,y(t),\sum_{k=0}^\infty \frac{y^{(k)}(t)}{k!}(-r)^k\right)$$ and it is known that DDE have characteristics that ODE have not.
• The function $f$ must be linear in $y(t)$ and $y(t-r)$ for this to apply (and additionally not every solution of the delay equation will be a solution of the equation with the Taylor expansion), so there some big restrictions on what DDE you can consider here. Do these restrictions still allow for phenomena incompatible with ODEs? Commented Sep 14, 2019 at 10:39