(a) Explanation Given information: The extended linearity principle for the nonhomogeneous equation is d y d t = a ( t ) y + b ( t ) Calculation: The extended linearity principle for the nonhomogeneous equation is, d y d t = a ( t ) y + b ( t ) -----(1) If y h ( t ) b e a s o l u t i o n o f a s s o c i a t e d hom o g e n o u s e q u a t i o n t h e n , d y h ( t ) d t = a ( t ) y h ( t ) − − − − ( 2 ) a n d i f , y p ( t ) b e a n y s o l u t i o n o f e q u a t i o n ( 1 ) t h e n , d y p ( t ) d t = a ( t ) y p ( t ) + b ( t ) − − − − ( 3 ) N o w t o p r o v e t h a t y h ( t ) + y p ( t ) i s a s o l u t i o n o f ( 1 ) (b) To determine To calculate: To prove that y p ( t ) − y q ( t ) is a solution to the associated homogenous equation d ( y p − y q ) d t .

4th Edition

ISBN: 9780495562009

Author: Blanchard Devaney Gl

Publisher: Cengage Learning

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Chapter 1.8, Problem 33E

(a)

To determine

To calculate:To prove that yh(t)+yp(t) satisfies the nonhomogeneous equation by calculating d(yh+yp)dt .

(b)

To determine

To calculate: To prove that yp(t)−yq(t) is a solution to the associated homogenous equation d(yp−yq)dt .

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Chapter 1.8, Problem 32E

Chapter 1.8, Problem 34E

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To prove that y h ( t ) + y p ( t ) satisfies the nonhomogeneous equation by calculating d ( y h + y p ) d t .

Solution Summary: The author explains the extended linearity principle for the nonhomogeneous equation.