The Wolfram Language's approach to differential operators provides both an elegant and a convenient representation of mathematical structures, and an immediate framework for strong algorithmic computation. With breakthrough methods developed at Wolfram Research, the Wolfram Language can perform direct symbolic manipulations on objects that represent solutions to differential equations.

Derivative — representation of derivatives with respect to any function argument

y', y'', etc. — short forms for Derivative

Grad () — gradient

Div ()  ▪  Curl ()  ▪  Laplacian ()

DSolve — solve systems of differential equations symbolically

NDSolve  ▪  DSolveValue  ▪  NDSolveValue  ▪  GreenFunction

DEigensystem — find eigenvalues and eigenfunctions for differential operators

DEigenvalues  ▪  NDEigensystem  ▪  NDEigenvalues

DifferentialRoot — symbolic representation of solutions to linear differential equations

FunctionExpand — expand differential roots in terms of special functions etc.

DifferentialRootReduce — canonicalize to differential root form