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part_diff_eqn

# part_diff_eqn - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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P. Partial Differential Equations An important application of the higher partial derivatives is that they are used in partial differential equations to express some laws of physics which are basic to most science and engineering subjects. In this section, we will give examples of a few such equations. The reason is partly cultural, so you meet these equations early and learn to recognize them, and partly technical: to give you a little more practice with the chain rule and computing higher derivatives. A partial differential equation, PDE for short, is an equation involving some unknown function of several variables and one or more of its partial derivatives. For example, is such an equation. Evidently here the unknown function is a function of two variables we infer this from the equation, since only x and y occur in it as independent variables. In general a solution of a partial differential equation is a differentiable function that satisfies it. In the above example, the functions w = xnyn any n all are solutions to the equation. In general, PDE's have many solutions, far too many to find all of them. The problem is always to find the one solution satisfying some extra
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part_diff_eqn - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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