Laplace / inverse laplace transform

In summary, Laplace transform is a mathematical operation used to transform a function of time into a function of complex frequency. Its inverse operation, inverse Laplace transform, allows us to retrieve the original function from its transformed form. Laplace transform is calculated by integrating the function multiplied by e^-st. There are two types of Laplace transform - one-sided and two-sided, with the latter being used for functions that are not zero for negative values of time. Laplace transform has many applications in various fields such as engineering, physics, and probability and statistics.
  • #1
goohu
54
3
View attachment 9453

Problem: Find a (limited?) solution to the diff eq.

View attachment 9454
At the end of the solution, when you transform \(\displaystyle \frac{-1}{s+1} + \frac{2}{s-3}\)
why doesn't it become \(\displaystyle -e^{-t} + 2e^{3t} \), t>0 ?
 

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  • #2
That would be correct if the right hand side of the equation were 0. But the delta functions on the right mean that those are correct only for x greater than certain values so the step functions are needed.
 
  • #3
What values are those and how are the step functions used?
 
  • #4
Do you know what "[tex]\delta(x)[/tex]" and "[tex]\delta'(x)[/tex]" mean?
 

Related to Laplace / inverse laplace transform

1. What is the Laplace transform?

The Laplace transform is a mathematical operation that transforms a function of time into a function of complex frequency, making it easier to solve differential equations. It is commonly used in engineering and physics to analyze systems and their behavior over time.

2. What is the inverse Laplace transform?

The inverse Laplace transform is the reverse operation of the Laplace transform, and it converts a function of complex frequency back into a function of time. It is used to find the original function from its transformed form.

3. How is the Laplace transform calculated?

The Laplace transform is calculated by integrating the function of time multiplied by the exponential function e^-st, where s is a complex variable. The result is a function of complex frequency, which can then be used to solve differential equations.

4. What is the region of convergence in the Laplace transform?

The region of convergence (ROC) is the range of values for the complex variable s for which the Laplace transform converges. It is an important concept in the Laplace transform as it determines the validity and accuracy of the transformed function.

5. What are the applications of the Laplace transform?

The Laplace transform has various applications in engineering, physics, and mathematics. It is used to solve differential equations, analyze control systems, and study the behavior of circuits and mechanical systems. It is also used in signal processing, image processing, and probability theory.

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