Explanation Given: The function is s 2 − 26 s − 47 ( s − 1 ) ( s + 2 ) ( s + 5 ) Approach: Suppose the function Q ( x ) can be factorized as Q ( x ) = ( a 1 x + b 1 ) ( a 2 x + b 2 ) ... ( a n x + b n ) Partial fraction expansion is given by P ( x ) Q ( x ) = A 1 a 1 x + b 1 + A 2 a 2 x + b 2 + ... + A n a n x + b n Calculation: Partial expansion of the given function is given by s 2 − 26 s − 47 ( s − 1 ) ( s + 2 ) ( s + 5 ) = A s − 1 + B s + 2 + C s + 5 = A ( s + 2 ) ( s + 5 ) + B ( s + 5 ) ( s − 1 ) + C ( s − 1 ) ( s + 2 ) ( s − 1 ) ( s + 2 ) ( s + 5 ) = A ( s 2 + 7 s + 10 ) + B ( s 2 + 4 s − 5 ) + C ( s 2 + s − 2 ) ( s − 1 ) ( s + 2 ) ( s + 5 ) = s 2 ( A + B + C ) + s ( 7 A + 4 B + C ) + 10 A − 5 B − 2 C ( s − 1 ) ( s + 2 ) ( s + 5 ) Multiply each side by the common denominator ( s − 1 ) ( s + 2 ) ( s + 5 ) s 2 − 26 s − 47 = s 2 ( A + B + C ) + s ( 7 A + 4 B + C ) + 10 A − 5 B − 2 C Compare both the sides A + B + C = 1 ( 1 ) 7 A + 4 B + C = − 26 ( 2 ) 10 A − 5 B − 2 C = − 47 ... ( 3 ) Multiply equation ( 1 ) by − 1 − A − B − C = − 1 ( 4 ) Add equation ( 2 ) and equation ( 4 ) 7 A + 4 B + C = − 26 + − A − B − C = − 1 _ 6 A + 3 B = − 27 6 A + 3 B = − 27

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 7.4, Problem 11E

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The partial fraction expansion of the rational function.

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Chapter 7.4, Problem 10E

Chapter 7.4, Problem 12E

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Fundamentals of Differential Equations and Boundary Value Problems

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The partial fraction expansion of the rational function.

Solution Summary: The author explains how the partial tion expansion of the rational function is given by rlls2-26s-47.

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To find: The partial fraction expansion of the rational function.

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To find:

The partial fraction expansion of the rational function.