Explanation Given: The given expression is L − 1 { s s 2 − 10 s + 29 } . Formula used: The formulae for the inverse Laplace transform are as follows, L − 1 { k s 2 − k 2 } = sinh k t L − 1 { s s 2 − k 2 } = cosh k t And if L − 1 { F ( s ) } = f ( t ) then, L − 1 { F ( s − a ) } = e a t f ( t ) Here a is any constant. Linearity property of the inverse Laplace transform is as follows, If a and b be any constant and f and g be any function of s then, L − 1 { a f ( s ) + b g ( s ) } = a L − 1 { f ( s ) } + b L − 1 { g ( s ) } Calculation: Use the property above to write the Laplace transform of the given function. L − 1 { s s 2 − 10 s + 29 } = L − 1 { s s 2 − 10 s + 25 + 4 } = L − 1 {

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 4, Problem 17CR

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To fill: The blank in the statement “L−1{s(s−5)2+22}=_______”.

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Chapter 4, Problem 16CR

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2. Consider the ODE u' = ƒ (u) = u² + r where r is a parameter that can take the values r = −1, −0.5, -0.1, 0.1. For each value of r: (a) Sketch ƒ(u) = u² + r and determine the equilibrium points. (b) Draw the phase line. (d) Determine the stability of the equilibrium points. (d) Plot the direction field and some sample solutions,i.e., u(t) (e) Describe how location of the equilibrium points and their stability change as you increase the parameter r. (f) Using the matlab program phaseline.m generate a solution for each value of r and the initial condition u(0) = 0.9. Print and turn in your result for r = −1. Do not forget to add a figure caption. (g) In the matlab program phaseline.m set the initial condition to u(0) = 1.1 and simulate the ode over the time interval t = [0, 10] for different values of r. What happens? Why? You do not need to turn in a plot for (g), just describe what happens.

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The blank in the statement “ L − 1 { s ( s − 5 ) 2 + 2 2 } = _ _ _ _ _ _ _ ”.

Solution Summary: The author explains the formulae for the inverse Laplace transform: lL-1lefts

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To fill: The blank in the statement “L−1{s(s−5)2+22}=_______”.

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Chapter 4 Solutions