What Is the Laplace Transform Formula for \(\mathcal{L}[f(t) \cdot g'(t)]\)?

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SUMMARY

The Laplace Transform formula for \(\mathcal{L}[f(t) \cdot g'(t)]\) is established as \(\mathcal{L}[f(t) \cdot g'(t)] = F(s)G(s) - f(0)G(s)\), where \(F(s)\) is the Laplace Transform of \(f(t)\) and \(G(s)\) is the Laplace Transform of \(g(t)\). This formula allows for the transformation of the product of a function and the derivative of another function without direct integration. The discussion emphasizes the utility of this formula in simplifying complex transformations in control theory and differential equations.

PREREQUISITES
  • Understanding of Laplace Transforms and their properties
  • Familiarity with functions and their derivatives
  • Basic knowledge of control theory concepts
  • Proficiency in mathematical notation and manipulation
NEXT STEPS
  • Study the properties of Laplace Transforms in detail
  • Explore applications of Laplace Transforms in solving differential equations
  • Learn about convolution theorem in the context of Laplace Transforms
  • Investigate the implications of the formula in control system design
USEFUL FOR

Mathematicians, engineers, and students in fields involving differential equations and control systems will benefit from this discussion, particularly those seeking efficient methods for applying Laplace Transforms.

HWGXX7
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Hello,

I'am looking for de correct transformation formule:\mathcal{L}[f(t).g'(t)]
(and proof).

I'am not looking for method to solve it by means of integrating g'(t), offcourse this a possible way. But assume that g(t) is much work to calculate.

So is there a good one to one formule for it?

ty&grtz
 
Physics news on Phys.org
Anyone an idea?

ty
 

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