Explanation Given that, the differential equation is, y ″ − 8 y ′ + 20 y = 100 x 2 − 26 x e x . In order to find the auxiliary equation, first compare the associate homogenous equation of differential equation. y ″ − 8 y ′ + 20 y Here, the auxiliary equation is m 2 − 8 m + 20 = 0 . Solve the auxiliary equation m 2 − 8 m + 20 = 0 for m as follows. m 2 − 8 m + 20 = 0 m 2 − 8 m + 20 = 0 Use quadratic formula and obtain the values of m as follows. m = − b ± b 2 − 4 a c 2 a m = − ( − 8 ) ± ( − 8 ) 2 − 4 ⋅ 1 ⋅ 20 2 ⋅ 1 = 8 ± 64 − 80 2 = 8 ± − 16 2 Simplify further, m = 8 ± − 16 2 = 2 ( 4 ± i 2 ) 2 = 4 ± i 2 Thus, the values of m are m 1 = 4 + 2 i and m 2 = 4 − 2 i . Hence, the complimentary function is y c = e 4 x ( c 1 cos 2 x + c 2 sin 2 x ) . Compare the differential equation y ″ − 8 y ′ + 20 y = 100 x 2 − 26 x e x with y ″ − 8 y ′ + 20 y = g ( x ) and obtain the function g ( x ) as, g ( x ) = 100 x 2 − 26 x e x . Assume that a particular solution of the given differential equation is also in the form of a quadratic function as, y p = A x 2 + B x + C + ( D x + E ) e x . Differentiate the function y p = A x 2 + B x + C + ( D x + E ) e x with respect to x as follows. y p = A x 2 + B x + C + ( D x + E ) e x y ′ p = 2 A x + B + 0 + D ( x e x + e x ) + E e x y ″ p = 2 A + 0 + D ( x e x + e x + e x ) + E e x y ″ p = 2 A + D x e x + ( 2 D + E ) e x Substitute y p = A x 2 + B x + C + ( D x + E ) e x , y ′ p = 2 A x + B + D x e x + ( D + E ) e x and y ″ p = 2 A + D x e x + ( 2 D + E ) e x in the differential equation y ″ − 8 y ′ + 20 y = 100 x 2 − 26 x e x and obtain the value of A , B , C , D and E as follows

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.4, Problem 6E

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To solve: The differential equation y″−8y′+20y=100x2−26xex by undetermined coefficient.

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Chapter 3.4, Problem 5E

Chapter 3.4, Problem 7E

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The differential equation y ″ − 8 y ′ + 20 y = 100 x 2 − 26 x e x by undetermined coefficient.

Solution Summary: The author explains how to solve the differential equation by undetermined coefficient. The auxiliary equation is m2-8m+20=0.

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To solve: The differential equation y″−8y′+20y=100x2−26xex by undetermined coefficient.

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Chapter 3 Solutions