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Consider the following initial value problem, representing the response of a damped oscillator subject to the discontinuous applied force f (t): S1 2

Consider the following initial value problem, representing the response of a damped oscillator subject to the discontinuous applied force f (t): S1 2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the following initial value problem, representing the response of a damped oscillator subject to the discontinuous applied force f (t): S1 2<t< 4, 0 otherwise. 3y" + 8 + 41y = f(t), y(0) = 6, (0) = 6, f(t) = In the following parts, use h(t c) for the Heaviside function h(t) when necessary. a. First, compute the Laplace transform of f (t). L{f(t)}(s) = b. Next, take the Laplace transform of the left-hand-side of the differential equation, set it equal to your answer from Part a. and solve for L{y(t)}. L{y(t)}(s) = c. We will need to take the inverse Laplace transform in order to find y(t). To do so, let's first rewrite L{y(t)} as *(;- P() -(:- P) + 6P(0) + 6G(») -2s -4s L{y(t)}(s) = 41 F(s) 41 where F(s) = and G(s) =
Transcribed Image Text:Consider the following initial value problem, representing the response of a damped oscillator subject to the discontinuous applied force f (t): S1 2<t< 4, 0 otherwise. 3y" + 8 + 41y = f(t), y(0) = 6, (0) = 6, f(t) = In the following parts, use h(t c) for the Heaviside function h(t) when necessary. a. First, compute the Laplace transform of f (t). L{f(t)}(s) = b. Next, take the Laplace transform of the left-hand-side of the differential equation, set it equal to your answer from Part a. and solve for L{y(t)}. L{y(t)}(s) = c. We will need to take the inverse Laplace transform in order to find y(t). To do so, let's first rewrite L{y(t)} as *(;- P() -(:- P) + 6P(0) + 6G(») -2s -4s L{y(t)}(s) = 41 F(s) 41 where F(s) = and G(s) =
d. Part c. indicates we will need L{F(s)} and L{G(s)}. Let's go ahead and find those now. -1 L-'{F(s)}(t) = and L-{G(s)}(t) = e. Use your answer in Part d. to compute L¯'{e-2$ F(s)} and L¬1{e-4$ F(s)}. L-l{e-"F(s)}(t) = and L{e-*F(s)}(t) = f. Finally, combine all the previous steps to write down y(t). y(t) =
Transcribed Image Text:d. Part c. indicates we will need L{F(s)} and L{G(s)}. Let's go ahead and find those now. -1 L-'{F(s)}(t) = and L-{G(s)}(t) = e. Use your answer in Part d. to compute L¯'{e-2$ F(s)} and L¬1{e-4$ F(s)}. L-l{e-"F(s)}(t) = and L{e-*F(s)}(t) = f. Finally, combine all the previous steps to write down y(t). y(t) =
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