Discontinuous

# Discontinuous - Discontinuous RHS The method of Laplace...

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Discontinuous RHS The method of Laplace transforms gives us a way of dealing with the nonhomogeneous equation y 00 + p ( x ) y 0 + q ( x ) y = g ( x ) when g ( x ) is piecewise continuous, which for our purposes we will take to mean a function that is continuous except for ﬁnitely many jump discontinuities. We will construct our piecewise continuous functions in terms of the following important function: Definition The piecewise continuous function H c ( x ) is deﬁned by the formula H c ( x ) = ± 0 0 x < c 1 x c This is called a Heaviside function after Oliver Heaviside. Now we can write piecewise continuous functions in terms of the Heaviside function as demonstrated in the following example. I Example Rewrite the piecewise continuous function g ( x ) = x 2 0 x < 3 x 3 x < 6 5 x 6 in terms of Heaviside functions. Solution First we will construct functions that are 1 on each interval of the intervals of deﬁnition appearing in g ( x ) and 0 oﬀ of these intervals. The function

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Discontinuous - Discontinuous RHS The method of Laplace...

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