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Theory 5. To solve differential equations with Laplace transforms, we use the fact that L{y"} = s²Y – sy(0) — s'(0). In this problem, we will derive and extend that result. (a) Use the integral definition of the Laplace transform: L {f(t)} = f(t)est dt for - formula to show that L{y'} = sL{y} − y(0). Hint: integration by parts. (b) Now, use part (a) and the fact that y" (y')' to derive the result. You should not be computing any more integrals here, just use the result of part (a) on y" and then again on y'. (c) = Might as well extend this. Derive a result for L{y"}. Hopefully, you see a pattern. We can use this to solve IVPs of any order.

Theory 5. To solve differential equations with Laplace transforms, we use the fact that L{y"} = s²Y – sy(0) — s'(0). In this problem, we will derive and extend that result. (a) Use the integral definition of the Laplace transform: L {f(t)} = f(t)est dt for - formula to show that L{y'} = sL{y} − y(0). Hint: integration by parts. (b) Now, use part (a) and the fact that y" (y')' to derive the result. You should not be computing any more integrals here, just use the result of part (a) on y" and then again on y'. (c) = Might as well extend this. Derive a result for L{y"}. Hopefully, you see a pattern. We can use this to solve IVPs of any order.

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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Theory 5. To solve differential equations with Laplace transforms, we use the fact that L{y"} = s²Y – sy(0) — s'(0). In this problem, we will derive and extend that result. (a) Use the integral definition of the Laplace transform: L {f(t)} = √ ° ° f(t)e=st dt (c) formula to show that L{y'} = sL{y} − y(0). Hint: integration by parts. (b) Now, use part (a) and the fact that y" = (y')' to derive the result. You should not be computing any more integrals here, just use the result of part (a) on y" and then again on y'. Might as well extend this. Derive a result for L{y""}. Hopefully, you see a pattern. We can use this to solve IVPs of any order.
Transcribed Image Text:Theory 5. To solve differential equations with Laplace transforms, we use the fact that L{y"} = s²Y – sy(0) — s'(0). In this problem, we will derive and extend that result. (a) Use the integral definition of the Laplace transform: L {f(t)} = √ ° ° f(t)e=st dt (c) formula to show that L{y'} = sL{y} − y(0). Hint: integration by parts. (b) Now, use part (a) and the fact that y" = (y')' to derive the result. You should not be computing any more integrals here, just use the result of part (a) on y" and then again on y'. Might as well extend this. Derive a result for L{y""}. Hopefully, you see a pattern. We can use this to solve IVPs of any order.
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