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Each of the examples of dierential equation we have

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Unformatted text preview: involve partial derivatives. Each of the examples of differential equation we have seen are types of ODEs, so here is an example of a PDE: Say that we have a rod of length L sitting in or above a heat source. (For example, the rod might have one end in a furnace.) We wish to compute the temperature u = u(x, t) as a position x (where 0 ≤ x ≤ L) and a time t. Note that both time t and position x are independent variables, and u = u(x, t) is a dependent variable. The differential equation which models this physical situation is called the Heat Equation : ∂2u ∂u = ∂ x2 ∂t for some nonzero constant α which depends on the characteristics of the rod. In this course, we will only be interested in ordinary differential equa...
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