DIFFERENTIAL EQUATIONS-NEXTGEN WILEYPLUS

3rd Edition

ISBN: 9781119764564

Author: BRANNAN

Publisher: WILEY

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Textbook Question

Chapter 5.6, Problem 17P

Modify the problem in Example 1 of this section by replacing the given forcing function g(t) by

$f ( t ) = u 5 ( t ) ( t − 5 ) − u 5 + k ( t ) ( t − 5 − k ) k .$

a) Sketch the graph of $f ( t )$ and describe how it depends on $k$ . For what value of $k$ is $f ( t )$ identical to $g ( t )$ in the example ?

b) Solve the initial value problem

$y ' ' + 4 y = f ( t ) ,$ $y ( 0 ) = 0 ,$ $y ' ( 0 ) = 0$

c) The solution in part (b) depends on $k$ , but for sufficiently large $t$ the solution is always a simple harmonic oscillation about $y = 1 4$ . Try to decide how the amplitude of this eventual oscillation depends on $k$ . Then confirm your conclusion by plotting the solution for a few different values of $k .$

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Chapter 5.6, Problem 16P

Chapter 5.6, Problem 18P

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DIFFERENTIAL EQUATIONS-NEXTGEN WILEYPLUS

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