a laplace miracle

a laplace miracle
Why does the Laplace transform work? Why do you use it to solve differential equations? It's all based on the following miracle: The Laplace transform turns derivatives into multiplication, and therefore differential equations. I prove a key identity related to the laplace transform, which will be crucial for solving initial value problems. This is a tool that is incredibly useful for physics and engineering as well
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[@SbF6H]

For me, the value of Laplace transform is in analysing the frequency responses of various circuits, filters, etc.

[@edmundwoolliams1240]

But then you have to do an inverse laplace transform afterwards, which requires complex analysis

[@YouTube_username_not_found]

And it turns linear PDEs into simpler linear PDEs (PDEs with n variables become PDEs with n-1 variables)

[@markgross9582]

Could you do a series on the justification for the laplace transform via distribution theory?

[@drpeyam]

A series?

[@MuPrimeMath]

So nice!

[@drpeyam]

Omg hi MuPrimeMath!! Hope all is well :)

[@phill3986]

Nice 👍🎉👏

[@redroach401]

Very good explanation! What is the laplace transform of Y(t) though?

[@GRosa]

That obviously depends on Y(t)

[@redroach401]

@GRosa no it doesn't, there is a generalized form in terms of Y(s)

[@ZacZounTheRamingo]

I have some difficulties solving the inverse Laplace transform of 1/s^b where 0

[@redroach401]

Go to a laplace transform table on the internet and locate your inverse laplace transform. Then isolate for s and calculate b to transform it into an equation with t.

[@prolarka]

Differentiation is not hard. Integration is.

[@drpeyam]

In analysis differentiation is harder than integration!