Laplace transform 2 | Laplace transform | Differential Equations | Khan Academy | Summary and Q&A

TL;DR

The Laplace transform of e to the at is 1/(s-a), assuming s is greater than a.

Key Insights

• ⌛ The Laplace transform is a mathematical technique that converts a function of time into a function of a complex variable.
• 😀 The Laplace transform of e to the at is 1/(s-a), assuming s is greater than a.
• ❓ The convergence of the Laplace transform depends on the relationship between a and s.

Transcript

Let's keep doing some Laplace transforms. For one, it's good to see where a lot of those Laplace transform tables you'll see later on actually come from, and it just makes you comfortable with the mathematics. Which is really just kind of your second semester calculus mathematics, but it makes you comfortable with the whole notion of what we're doi... Read More

Questions & Answers

Q: What is the Laplace transform and how is it defined?

The Laplace transform is a mathematical tool used to convert a function of time into a function of a complex variable, s. It is defined as the improper integral of e^(-st) times f(t) with respect to dt.

Q: How does the Laplace transform of e to the at simplify?

The Laplace transform of e to the at is equal to 1/(s-a). This simplification occurs by substituting e to the at into the definition of the Laplace transform and integrating.

Q: Under what conditions does the Laplace transform of e to the at converge?

The Laplace transform of e to the at converges when a minus s is less than 0, or when a is less than s. In this case, the result is 1/(s-a).

Q: How does the Laplace transform of e to the at relate to the Laplace transform of 1?

The Laplace transform of e to the at is consistent with the Laplace transform of 1, as both result in the expression 1/(s-a). This shows that the Laplace transform is a consistent mathematical tool.

Summary & Key Takeaways

• The Laplace transform is a useful tool in mathematics and can be used to solve differential equations.

• The Laplace transform of a function f(t) is defined as the improper integral of e^(-st) times f(t) with respect to dt, where s is a constant.

• The Laplace transform of e to the at is equal to 1/(s-a), with the assumption that s is greater than a.