Laplace Transform of (1 - e^t + 2e^(-3t))cos(6t) using the First Translation Theorem

TL;DR

This content explains how to find the Laplace transform solution for a cosine function by applying formulas and shifts.

Transcript

in this problem we have to find the Laplace transform solution we'll start by distributing the cosine function to each of these terms here so first we have the Laplace of well one times cosine that's just going to be cosine 60 then - applause of e to the T times cosine of 60 and then plus and we can pull the two out so - applause transform of e to ... Read More

Key Insights

• ❓ The Laplace transform solution for a cosine function involves applying specific formulas and shifts.
• ❓ The Laplace transform of cosine KT is S/(S^2 + K^2).
• 💦 The Laplace transform of e^AT times a function involves dropping the exponential function and shifting the Laplace variable.
• 🧑‍🏭 The Laplace transform of cosine 60 is S/(S^2 + 36), considering the S factor in the cosine formula.
• ❓ Shifting occurs in the Laplace transform solution when there is an exponential function.
• 🇸🇹 The Laplace transform of e^-3T cosine 60 involves shifting from S to S + 3, resulting in (S + 3)/(S^2 + 36).
• ❓ The Laplace transform solution requires substituting the Laplace variables with their shifted values for accurate calculations.

Questions & Answers

Q: What is the first formula required to solve this problem?

The first formula needed is the Laplace transform of the cosine of KT, which states that if you have the Laplace of cosine KT, the Laplace variable is replaced by S, and it becomes S/(S^2 + K^2).

Q: What is the second formula required to solve this problem?

The second formula is the first translation theorem, which states that if you have the Laplace of e^AT times a function, you can drop the exponential function, take the Laplace of F, and then shift the Laplace variable from S to S - A.

Q: How do you calculate the Laplace transform of cosine 60?

The Laplace transform of cosine 60 is simply S/(S^2 + 36), as cosine has an S in its formula, and the square term is the sum of the squares of the cosine coefficient.

Q: How do you calculate the Laplace transform of e^-3T cosine 60?

To calculate the Laplace transform of e^-3T cosine 60, you perform a shift from S to S + 3, resulting in (S + 3)/(S^2 + 36) due to the Laplace transform properties.

Summary & Key Takeaways

• The content explains the steps to find the Laplace transform solution for a cosine function.

• The formulas used include the Laplace transform of a cosine function and the first translation theorem.

• The calculations involve distributing the cosine function, applying the Laplace transform formulas, and performing shifts.