Inverse Laplace Transform, Sect 7.4#35  Summary and Q&A
TL;DR
This video explains how to find the inverse Laplace transform using derivatives and breaks down the steps involved.
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.