DifferentialRoot

A holonomic (D-finite) function object — a linear ODE with polynomial(x) coefficients plus enough derivatives at one point, exactly as DifferentialRootReduce emits it. Applied to a real number, it sums the ODE's Taylor series about the stored expansion point numerically; applied to anything it can't sum (a symbolic point, or one outside the radius of convergence), it stays inert. Provided by @enumeratio/analytic.

Wolfram LanguageDifferentialRoot↗

DifferentialRoot(fn)(x0)the function's value at $x_0$, by summing the Taylor series the ODE (and the stored initial derivatives) determine.

Typed signatures

Type	Package	Applies
(any) -> any	analytic	always

Domain: Special functions

Details

• Ordinarily built by DifferentialRootReduce rather than by hand — the example below constructs one directly (the defining ODE of $e^x$) to show the evaluation contract on its own.
• Numeric only, unlike DifferenceRoot's exact rational arithmetic — this is the one place this package computes a numeric-only answer by design (see difference-root.ts's header).
• Attached to the native Apply operator the same way DifferenceRoot is (never redeclares it), so only a call whose head is DifferentialRoot(...) is intercepted.
• Declines when the expansion point is not an ordinary point of the ODE (the leading coefficient vanishes there), or the series hasn't visibly settled within a generous term budget — most often because the target is outside the radius of convergence.

Examples

Basic #

$y' = y$, $y(0){=}1$: $y(2) = e^2$

live ↗

In$\mathrm{DifferentialRoot}((y, x)↦\begin{cases}y^{\prime}(x)-y(x)=0\\y(0)=1\end{cases})\lhd2$Out$7.389\,056\,098\,930\,649$

About

Examples	1
Crosswalk	Wolfram Language

See also: DifferentialRootReduce, DifferenceRoot