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Notes 9F (Heavisides's cover-up method)

# Notes 9F (Heavisides's cover-up method) - MIT...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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F. HEAVISIDE'S COVER-UP METHOD The eponymous method was introduced by Oliver Heaviside as a fast way to do a decom- position into partial fractions. In 18.01 we need the partial fractions decomposition in order to integrate'rational functions (i.e., quotients of polynomials). In 18.03, it will be needed as an essential step in using the Laplace transform to solve differential equations, and in fact this ývas more or less Heaviside's original motivation. The cover-up method can be used to make a partial fractions decomposition of a rational function p-) whenever the denominator can be factored into distinct linearfactors. We illustrate with an example; though simple, it should convince you that the method is worth learning. x-7 Example 1. Decompose ( 1)( +2) into partial fractions. Solution. We know the answer will have the form x-7 A B (x-1)(x+2)- x-1 +x+2 To determine A by the cover-up method, on the left-hand side we mentally remove (or cover up with a finger) the factor x - 1 associated with A, and substitute x = 1 into what's left; this gives A: x-7 1-7 (2) (( + 2)x,=1 = - = -2= A. 1+2 Similarly, B is found by covering up the factor x + 2 on the left, and substituting x = -2 into what's left. This gives x-7 -2-7 x - = 3 = B. (x
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Notes 9F (Heavisides's cover-up method) - MIT...

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