Explanation Given information: Consider the nonhomogeneous linear equation d y d t + α ( t ) y = b 1 ( t ) + b 2 ( t ) . Here, b(t) is the sum of two functions and y h (t) is the solution of associated homogeneous equation d y d t + α ( t ) y = 0 , y 1 (t) is the solution of the equation d y d t + α ( t ) y = b 1 ( t ) and y 2 (t) is the solution of the equation d y d t + α ( t ) y = b 2 ( t ) . Calculation: The nonhomogeneous linear differential equation, d y d t + α ( t ) y = b 1 ( t ) + b 2 ( t ) ------- (1) Let y h (t) is the solution of the differential equation , d y d t + α ( t ) y = 0 ------ (2) d y h ( t ) d t + α ( t ) y h ( t ) = 0 Let y 1 (t) is the solution of the differential equation, d y d t + α ( t ) y = b 1 ( t ) ------ (3) d y 1 ( t ) d t + α ( t ) y 1 ( t ) = b 1 ( t ) Let y 2 (t) is the solution of the differential equation, d y d t + α ( t ) y = b 2 ( t ) ---- (4) d y 2 ( t ) d t + α ( t ) y 2 ( t ) = b 2 ( t ) Take the left hand side of the equation (1) and substitute as, y ( t ) = y h ( t ) + y 1 ( t ) + y 2

4th Edition

ISBN: 9780495561989

Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall

Publisher: Cengage Learning

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Chapter 1.8, Problem 19E

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To calculate:To prove that yh(t)+y1(t)+y2(t) is the solution of nonhomogeneous equation dydt+α(t)y=b1(t)+b2(t) .

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Chapter 1.8, Problem 18E

Chapter 1.8, Problem 20E

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To prove that y h ( t ) + y 1 ( t ) + y 2 ( t ) is the solution of nonhomogeneous equation d y d t + α ( t ) y = b 1 ( t ) + b 2 ( t ) .

Solution Summary: The author calculates the nonhomogeneous linear differential equation dyt+alpha (t)y=b_1(t).

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To calculate:To prove that yh(t)+y1(t)+y2(t) is the solution of nonhomogeneous equation dydt+α(t)y=b1(t)+b2(t) .

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Chapter 1 Solutions