# Introduction to Partial Differential Equations (PDEs)

A partial differential equation (PDE) is a relationship between an unknown function and its derivatives with respect to the variables .

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PDEs occur naturally in applications; they model the rate of change of a physical quantity with respect to both space variables and time variables. At this stage of development, DSolve typically only works with PDEs having two independent variables.

The order of a PDE is the order of the highest derivative that occurs in it. The previous equation is a first-order PDE.

A function is a *solution* to a given PDE if and its derivatives satisfy the equation.

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Here are some well-known examples of PDEs (clicking a link in the table will bring up the relevant examples). DSolve gives symbolic solutions to equations of all these types, with certain restrictions, particularly for second-order PDEs.

name of equation | general form | classification |

transport equation | with constant | linear first-order PDE |

Burgers' equation | quasilinear first-order PDE | |

eikonal equation | nonlinear first-order PDE | |

Laplace's equation | elliptic linear second-order PDE | |

wave equation | where is the speed of light | hyperbolic linear second-order PDE |

heat equation | where is the thermal diffusivity | parabolic linear second-order PDE |

Recall that the general solutions to PDEs involve arbitrary *functions* rather than arbitrary *constants*. The reason for this can be seen from the following example.

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If there are several arbitrary functions in the solution, they are labeled as C[1], C[2], and so on.