Inverse Laplace Transform, Sect 7.4#35

TL;DR

This video explains how to find the inverse Laplace transform using derivatives and breaks down the steps involved.

Transcript

okay we're going to figure the inverse La plus transform of Ln of s² + 9 over s² + 1 well you know the deal this is just like the previous video and you can watch that for the detail explanation right here first of all I'm going to use the LM property to break this apart so we will have the inverse La plus transform of Ln of s² + 9 and then minus L... Read More

Key Insights

• 😒 The video showcases the use of the property of Laplace transforms and differentiation to find the inverse Laplace transform.
• 😑 Differentiation inside the Laplace transform allows for manipulating the expression and obtaining simpler forms.
• 😑 Factoring the expression into the perfect square form facilitates the computation of the inverse Laplace transform.

Questions & Answers

Q: How is the inverse Laplace transform of Ln(s² + 9) - Ln(s² + 1) computed?

The video explains that the expression is first differentiated inside the Laplace transform. The derivative of Ln(s² + 9) is 2s/(s² + 9), and the derivative of Ln(s² + 1) is 2s/(s² + 1). These derivatives are then multiplied by -1/T and factored to obtain the final result.

Q: What is the significance of factoring the expression?

Factoring the expression helps simplify it into the perfect square form, which allows for easier computation of the inverse Laplace transform. In this case, the factored expression results in -2cos(3T) + 2cos(T)/T.

Q: Why is it necessary to differentiate inside the Laplace transform?

Differentiating inside the Laplace transform helps manipulate the expression and obtain a simpler form that can be easily inverted. It allows for the application of known inverse Laplace transforms of basic functions.

Q: How is the final answer derived from the factored expression?

The final answer is obtained by distributing the factors and simplifying the expression. The result is -2cos(3T) + 2cos(T)/T.

Summary & Key Takeaways

• The video demonstrates how to find the inverse Laplace transform of Ln(s² + 9) - Ln(s² + 1) using the property of Laplace transforms.

• To simplify the expression, the derivatives are taken inside the Laplace transform and multiplied by -1/T before computing the inverse Laplace transform.

• After differentiating, the expression is factored and rewritten in the perfect square form to obtain the final result of -2cos(3T) + 2cos(T)/T.