Question

I know the basics of using the Laplace transform in finding solutions to differential equations today, but is there anything online that relates to how Laplace used the concept of integral transforms to determine solutions to his problems?

Answer 1

im prolly reading to much into my writing assignment; and they prolly just want a demonstartion of how we apply the LT today ... but thats not how its worded :)

Answer 2

Do a search of Laplace and his mathematics in particular on his work on diffyQs.

Answer 3

Blasphemy!

As a pure mathematics advocate, I will abstain from answering this question. ;P

Answer 4

Perhaps Laplace never used his concepts to solve his problems and merely derived them out of pure interest, pleasure and amusement.

Answer 5

lets start a riot!! :)

this is what ive written so far

The search on Laplace has portrayed him as a rather arrogant, egotistical, narcissist. Yet, it appears that he had the wherewithal to back up such an attitude. Laplace created many new mathematical methods, but his motivation with regards towards mathematics rested in its capacity to help him study nature. In his acclaimed work, Traite de Mechanique Celeste, Laplace used calculus and differential equations, along with Newton’s theories pertaining to gravity and celestial motions, to solve many of the issues that had been left open by the emerging theories of his day. Laplace is known for his founding work in probability theory as well. His book entitled, Theorie Analytique des Probabilites, includes some basic applications of what we today refer to as the Laplace transform. The Laplace transform is one of the oldest integral transforms that have been used in finding solutions to linear differential equations and integral equations. Euler is actually credited with introducing integral transforms; but, it was Spitzer who attached the name “Laplace” to the expression: y=abesx fs ds. When used appropriately, the Laplace transform reduces the problem of finding a solution to a differential equation by literally transforming the diffyQ into an algebraic expression. Today, the Laplace transform usually takes a form similar to:

F(s)=L(f(t))= int[0,∞) e-st f(t)dt

With some restrictions to the values that we can use for s this structure transforms f(t) into a function of s. The Laplace transform has the attribute that it creates a unique solution that can be inverted to obtain a suitable solution to the given equations.

Answer 6

Laplace wasnt the pure math type; he only used math as a means to learn his lovely applied stuff

Answer 7

That is very well written, but I do not think that it would really add that much more weight to it to list Laplace's applications of his transform. Perhaps you should contrast it with a different transform, such as Fourier's.

Answer 8

Well, i feel that would take me off the topic of the writing assignment ....

Answer 9

going off topic is something that comes natural to me tho ;)

Answer 10

It wouldn't if you do it briefly, and it would let your readers (professor) know that you are well aware of the existence of other (better ;P) methods.

Answer 11

ill consider it. i might even opt for a doodle of an oak tree during sunset to fill space