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2. Try again You have answered 5 out of 9 parts correctly. Consider the following initial value problem, representing the response of a damped oscillator subject to the discontinuous applied force f(t): Si 2

2. Try again You have answered 5 out of 9 parts correctly. Consider the following initial value problem, representing the response of a damped oscillator subject to the discontinuous applied force f(t): Si 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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2. Try again You have answered 5 out of 9 parts correctly. Consider the following initial value problem, representing the response of a damped oscillator subject to the discontinuous applied force f(t): Si 2<t< 4, 0 otherwise. {o y" + 8y/ + 41y = f(t), y(0) = 6, y(0) = 6, f(t) = In the following parts, use h(t – c) for the Heaviside function he(t) when necessary. a. First, compute the Laplace transform of f(t). -2s 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)}. -2s -4s e As) L{y(t)}(s) = + 6s + 54 (s2+ 8s +41 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 LiwO}{O) = (; -10) -(-10)- -2s 4s f (s) f (s) ) + 6f (s) + 6G(s) where (s+8) (3? + 1 f (s) = + 8s +41) and G(s) = (s² + 8s+41) d. Part c. indicates we will need L{f (s)} and L{G(s)}. Let's go ahead and find those now. L-{f (s)}(t) = and L-{G(s)}(t) = e -4t sin(5t) Ins Windows Ink Wor
Transcribed Image Text:2. Try again You have answered 5 out of 9 parts correctly. Consider the following initial value problem, representing the response of a damped oscillator subject to the discontinuous applied force f(t): Si 2<t< 4, 0 otherwise. {o y" + 8y/ + 41y = f(t), y(0) = 6, y(0) = 6, f(t) = In the following parts, use h(t – c) for the Heaviside function he(t) when necessary. a. First, compute the Laplace transform of f(t). -2s 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)}. -2s -4s e As) L{y(t)}(s) = + 6s + 54 (s2+ 8s +41 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 LiwO}{O) = (; -10) -(-10)- -2s 4s f (s) f (s) ) + 6f (s) + 6G(s) where (s+8) (3? + 1 f (s) = + 8s +41) and G(s) = (s² + 8s+41) d. Part c. indicates we will need L{f (s)} and L{G(s)}. Let's go ahead and find those now. L-{f (s)}(t) = and L-{G(s)}(t) = e -4t sin(5t) Ins Windows Ink Wor
e. Use your answer in Part d. to compute L{e-2°f (s)} and L{e 4f (s)}. L-1{e-* f (s)}(t) = and L-{e-4f (s)}(t) = -2s f. Finally, combine all the previous steps to write down y(t). y(t)
Transcribed Image Text:e. Use your answer in Part d. to compute L{e-2°f (s)} and L{e 4f (s)}. L-1{e-* f (s)}(t) = and L-{e-4f (s)}(t) = -2s f. Finally, combine all the previous steps to write down y(t). y(t)
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