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first seen date

2024-10-12 08:26:23

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Open Graph

title

Pengfei's Notes

description

site name

Pengfei's Notes

author

updated

2026-01-11 11:13:00

raw text

Pengfei's Notes Pengfei's Notes Solve IVPs using Laplace Transform – a problematic example Laplace transform has been used to solve Initial Value Problems. This is a topic covered in a standard differential equation course at most universities. In this post I will describe a strange situation: the solution of 2nd order ODE obtained by using Laplace transform appears to violate one of the two initial conditions. Then a short justification will be given. It turns out to be a very simple problem, just one may get caught off guard. Background . Consider a 2nd order linear constant coefficient differential equation: , , . Applying Laplace transform, we get , where the capital-case functions are the Laplace transforms of the corresponding lowercase functions: and . It follows that , or equally, . Applying the inverse transform, we obtain the solution of the IVP. Laplace Transform of the delta functions . Let be the delta function at . We know that . In part...

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