Automatically selecting between hundreds of powerful and in many cases original algorithms, the Wolfram Language provides both numerical and symbolic solving of differential equations (ODEs, PDEs, DAEs, DDEs, ...). With equations conveniently specified symbolically, the Wolfram Language uses both its rich set of special functions and its unique symbolic interpolating functions to represent solutions in forms that can immediately be manipulated or visualized.

y'[x] (Derivative) — derivative of a function

DSolve — symbolic solution to differential equations

DSolveValue — find an expression for the symbolic solution of a differential equation

NDSolve — numerical solution to differential equations

InterpolatingFunction — interpolating function used in solutions

ParametricNDSolveValue — numerical solution to differential equations with parameters

NDSolveValue  ▪  ParametricNDSolve  ▪  ParametricFunction

Differential Equations with Events »

WhenEvent — actions to be taken whenever an event occurs in a differential equation

Partial Differential Equations »

DirichletCondition — specify Dirichlet conditions for partial differential equations

NeumannValue — specify Neumann and Robin conditions

PeriodicBoundaryCondition — specify periodic boundary conditions

D  ▪  Grad  ▪  Div  ▪  Curl  ▪  Laplacian  ▪  ...

Differential Eigen Problems

NDEigensystem — numerical eigenvalues and eigenfunctions from a differential equation

NDEigenvalues — numerical eigenvalues from a differential equation

DEigensystem — symbolic eigenvalues and eigenfunctions from differential equations

DEigenvalues — symbolic eigenvalues from a differential equation

Stability Analysis

DFixedPoints — fixed points for a system of differential equations

DStabilityConditions — stability conditions for a system of differential equations

Simulations

NBodySimulation — simulation of idealized n-body systems

SystemModelSimulate — simulate a wide range of system models

Options

AccuracyGoal  ▪  PrecisionGoal  ▪  WorkingPrecision

Method — select and tune many possible solver algorithms

StepMonitor, EvaluationMonitor — monitor the progress of a solution

Method Functions

GreenFunction — Green's function for a differential equation

CompleteIntegral — complete integral for a first-order partial differential equation

Wronskian — test linear independence of functions or ODE solutions

Orthogonalize  ▪  Normalize

Differential Functions »

DifferentialRoot — representation of solutions to linear differential equations

Visualization »

Plot  ▪  StreamPlot  ▪  VectorPlot