Laplace Transform of cosh^2(kt)

TL;DR

This video explains how to find the Laplace transform of the hyperbolic cosine squared function and provides step-by-step instructions.

Transcript

okay so we have to find the laplace transform of the hyperbolic cosine squared of kt so to do this we're going to use the definition of the hyperbolic cosine so recall that the hyperbolic cosine of say x is equal to well it's the average of e to the x and e to the negative x so it's e to the x plus e to the negative x it's all divided by 2. so in t... Read More

Key Insights

• 🗂️ The hyperbolic cosine function can be defined as the average of e^x and e^-x divided by 2.
• ❎ When the hyperbolic cosine function is squared, it can be expanded using the formula for multiplying out the square of a sum.
• ❓ The Laplace transform of e^at is given by 1/(s-a), and the Laplace transform of 1 is 1/s.
• 😑 The Laplace transform is a linear operator, allowing us to apply it separately to each term in an expression.

Questions & Answers

Q: What is the definition of the hyperbolic cosine function?

The hyperbolic cosine function is the average of e^x and e^-x, divided by 2.

Q: How do we find the hyperbolic cosine squared function?

To find the hyperbolic cosine squared function, we square the hyperbolic cosine expression by using the formula for multiplying out the square of a sum.

Q: What is the formula for multiplying out the square of a sum?

The formula for multiplying out the square of a sum states that (a + b)^2 = a^2 + 2ab + b^2, where a and b are variables.

Q: How do we apply the Laplace transform to the hyperbolic cosine squared function?

We can apply the Laplace transform to each term separately and use the linearity property of the Laplace transform.

Summary & Key Takeaways

• The hyperbolic cosine function is defined as the average of e^x and e^-x divided by 2.

• To find the hyperbolic cosine squared function, we square the hyperbolic cosine expression.

• By applying the formula for multiplying out the square of a sum, we expand the squared expression.