IntroductionA partial differential equation (PDE) is a relationship between an unknown function  and its derivatives with respect to the variables  . Here is an example of a PDE. In[244]:=  |
PDEs occur naturally in applications because one tries to model the rate of change of a physical quantity with respect to both space variables and time variables. Here, only the case with two independent variables called x and y is considered. 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 u and its derivatives satisfy the equation. Here is one solution to the previous equation. In[245]:=  |
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This verifies the solution. In[246]:=  |
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Here are some well-known examples of PDEs. The terminology used in their classification is explained later in this section. DSolve gives symbolic solutions to equations of all these types, with certain restrictions, particularly for second-order PDEs.
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. The partial derivative with respect to y does not appear in this example, so an arbitrary function C[1][y] can be added to the solution, since the partial derivative of C[1][y] with respect to x is 0. In[247]:=  |
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If there are several arbitrary functions in the solution, they are labelled as C[1], C[2], and so on.
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