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Unformatted text preview: three equations is homogeneous linear , that is each term contains u or one of its derivatives to the ﬁrst power. This ensures that the principle
of superposition will hold,
u1 and u2 solutions ⇒ c1 u1 + c2 u2 is a solution, for any choice of constants c1 and c2 . The principle of superposition is essential
if we want to apply separation of variables and Fourier analysis techniques.
In the ﬁrst few sections of this chapter, we will derive these partial diﬀerential equations in several physical contexts. We will begin by using the divergence
115 theorem to derive the heat equation, which in turn reduces in the steadystate
case to Laplace’s equation. We then present two derivations of the wave equation, one for vibrating membranes and one for sound waves. Exactly the same
wave equation also describes electromagnetic waves, gravitational waves, or water waves in a linear approximation. It is remarkable that the principles developed to solve the three basic linear partial diﬀerential equations can be applied
in so many contexts.
In a few cases, it is p...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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