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bassplayer142

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- Thread starter bassplayer142
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In summary, there is a clear difference between partial and regular differential equations, with partial differential equations having an infinite number of states while regular differential equations have a finite number of states. This can restrict understanding in physics, as some equations may require partial differential equations to solve. However, some techniques such as Cauchy-Euler, substitution, and integrating factor may still work for both types of equations. The wave equation can be solved using separation of variables, but not all equations can be solved using regular differential equations, as there are conditions that must be satisfied for their existence. Some systems may be better represented using difference equations instead of differential equations.

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bassplayer142

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John Creighto

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bassplayer142

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John Creighto

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Well, the wave equation can be solved via separation of variables:

http://en.wikipedia.org/wiki/Separable_partial_differential_equation

http://www.math.ubc.ca/~feldman/m267/separation.pdf

http://en.wikipedia.org/wiki/Separable_partial_differential_equation

http://www.math.ubc.ca/~feldman/m267/separation.pdf

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bassplayer142

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John Creighto

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http://en.wikipedia.org/wiki/Picard-Lindelöf_theorem

Now if a solution exists, solving it is another matter. Some systems such as discrete systems are more practical to represented in terms of difference equations then differential equations.

Differential equations involve functions of one or more independent variables and their derivatives, while partial differential equations involve functions of multiple independent variables and their partial derivatives.

Differential and partial differential equations are used in many fields of science and engineering to model and analyze various phenomena, such as population growth, heat transfer, and fluid dynamics.

There are various techniques for solving differential and partial differential equations, including separation of variables, the method of characteristics, and numerical methods such as finite differences or finite elements.

Initial conditions specify the values of the dependent variable and its derivatives at a given starting point, while boundary conditions specify the behavior of the dependent variable at the boundaries of the domain. These conditions are essential for finding a unique solution to a differential or partial differential equation.

Yes, differential and partial differential equations can be used to make predictions about the behavior of a system over time. However, the accuracy of these predictions depends on the accuracy of the model and the quality of the initial and boundary conditions provided.

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