Mathematica'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, Mathematica 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
DSolve — solve systems of differential equations
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