# Defining a function through an ODE containing unspecified operators

I want to do some algebra using a function only defined through a DE containing unspecified operators. The DE is $$\partial_zu(z) = \left[\hat{D}+\hat{N}(z,u)\right] u(z).$$ Here $$u$$ lives in some function space (it is bounded, integrable, continuously differentiable,...) and $$\hat{D},\hat{N}$$ are bounded operators on said function space that don't necessarilly commute and only $$\hat{D}$$ is linear. How do I define $$u$$ in Mathematica?

My reason for asking this question: I derived that for $$u_I(z):=e^{-(z-z')\hat{D}}\cdot u(z)$$ that \begin{align} \partial_zu_I(z) &= \hat{N}_I(z,u_I)\cdot u_I(z),\\ \hat{N}_I(z,u_I) :&= e^{-(z-z')\hat{D}}\cdot \hat{N}(z,u_I)\cdot e^{(z-z')\hat{D}}. \end{align} I was wondering how to do such derivations with Mathematica.

• Do you want a numerical (approximate) solver or a symbolic (exact) solver? – Michael E2 May 8 at 16:49
• In DifferentialRoot one finds embedded in examples, the use of Function to code an operator. – Michael E2 May 8 at 16:52
• This is all symbolic. Nothing is numerically specified, so nothing numerical is even an option. It's purely about rewriting, using algebra/calculus. – Henk May 9 at 9:39