Explanation Given: The given initial value problem is y ″ − 9 y = g ( t ) ; y ( 0 ) = 2 , y ′ ( 0 ) = 0 Approach: 1) The transfer function of a linear system is defined as the ratio of the Laplace transform of the output function y ( t ) to the Laplace transform of the input function g ( t ) , under the assumption that all initial conditions are zero. That is, H ( s ) = Y ( s ) / G ( s ) . If the linear system is governed by the differential equation a y ″ + b y ′ + c y = g ( t ) , t > 0 Where a , b , and c are constants. Applying the Laplace transform and then assuming all initial conditions are zero, the equation reduces to ( a s 2 + b s + c ) ( Y ( s ) ) = G ( s ) . Thus, the transfer function for the differential equation is H ( s ) = Y ( s ) G ( s ) = 1 a s 2 + b s + c 2) The impulse response function is given by h ( t ) = L − 1 { H } ( t ) 3) The formula for the solution to the given initial value problem is given by, ( h ∗ g ) ( t ) + y k ( t ) . 4) L { e a t } = 1 s − a Calculation: Consider the initial value problem is, y ″ − 9 y = g ( t ) ; y ( 0 ) = 2 , y ′ ( 0 ) = 0 Compare the above differential equation with the equation a y ″ + b y ′ + c y = g ( t ) . So, this gives a = 1 , b = 0 , c = − 9 Thus, the transfer function of the system is H ( s ) = 1 a s 2 + b s + c = 1 ( 1 ) s 2 + 0 ( s ) − 9 = 1 s 2 − 9 Therefore the transfer functions is H ( s ) = 1 s 2 + 9 . And the impulse response function is h ( t ) = L − 1 { H } ( t ) = L − 1 { 1 s 2 − 9 } ( t ) = L − 1 { 1 ( s + 3 ) ( s − 3 ) } ( t ) = 1 6 L − 1 { ( s + 3 ) − ( s − 3 ) ( s + 3 ) ( s − 3 ) } ( t ) Further simplify the above equation as, h ( t ) = 1 6 L − 1 { ( s + 3 ) − ( s − 3 ) ( s + 3 ) ( s − 3 ) } ( t ) = 1 6 L − 1 { ( s + 3 ) ( s + 3 ) ( s − 3 ) } ( t ) − 1 6 L − 1 { ( s − 3 ) ( s + 3 ) ( s − 3 ) } ( t ) = 1 6 L − 1 { 1 ( s − 3 ) } ( t ) − 1 6 L − 1 { 1 ( s + 3 ) } ( t ) = 1 6 e 3 t − 1 6 e − 3 t Since, 1 2 ( e x − e − x ) = sinh ( x ) So,

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 7.8, Problem 24E

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The transfer function H(s) for the given system and the impulse response function h(t) and the formula for the solution to the given initial value problem.

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Chapter 7.8, Problem 23E

Chapter 7.8, Problem 25E

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The transfer function H ( s ) for the given system and the impulse response function h ( t ) and the formula for the solution to the given initial value problem.

Solution Summary: The author explains the transfer function of a linear system and the impulse response function for the given initial value problem.

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The transfer function H(s) for the given system and the impulse response function h(t) and the formula for the solution to the given initial value problem.

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The transfer function H ( s ) for the given system and the impulse response function h ( t ) and the formula for the solution to the given initial value problem.

Solution Summary: The author explains the transfer function of a linear system and the impulse response function for the given initial value problem.

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To find: The transfer function H(s) for the given system and the impulse response function h(t) and the formula for the solution to the given initial value problem.

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Chapter 7 Solutions

To find:

The transfer function H(s) for the given system and the impulse response function h(t) and the formula for the solution to the given initial value problem.