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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.
