Problem 5 Based on Multiplication By 't' Property - Laplace Transform - Engineering Mathematics 3

TL;DR

The video explains how to find the Laplace transform of t times j naught of a t by using the multiplication by t property and the change of scale property.

Transcript

so here we have to find out the laplace of into j naught of a t and given is the laplace of j naught of t is 1 upon square root of 1 plus s square now guys if you see the question it is laplace of t into some function so it is matching with a multiplication by t property so let me show you multiplication by t property looks like this we have a func... Read More

Key Insights

• ⌛ The Laplace transform of t times a function can be found using the multiplication by t property.
• ⚖️ The change of scale property allows for the derivation of the Laplace transform of a scaled function.
• 😃 The value of j naught of a t can be obtained from j naught of t using the change of scale property.
• ❎ The Laplace transform of j naught of a t is 1 upon square root of a square plus s square.
• ❎ The Laplace transform of t times j naught of a t is -s divided by (s square plus a square) raised to 3/2.
• 🎮 The video provides step-by-step explanations of the derivations and properties used.

Questions & Answers

Q: How is the value of Laplace transform of j naught of a t obtained from j naught of t?

The change of scale property is applied, which states that if Laplace transform of f of t is equal to phi of s, then the Laplace transform of f of a t is given by 1 upon a times phi of s upon a. By replacing s with s upon a in the function phi of s, the Laplace transform of j naught of a t is derived.

Q: What is the multiplication by t property?

The multiplication by t property states that if f of t is a function multiplied by t, then the Laplace transform of t times f of t is equal to -d by ds of phi of s, where phi of s is the Laplace transform of f of t.

Q: What is the final result for the Laplace transform of t times j naught of a t?

The Laplace transform is -s divided by (s square plus a square) raised to 3/2.

Q: How are the properties of change of scale and multiplication by t applied in finding the Laplace transform?

The change of scale property is used to obtain the value of j naught of a t from j naught of t. Then, the multiplication by t property is applied to find the Laplace transform of t times j naught of a t by taking the derivative of the Laplace transform function.

Summary & Key Takeaways

• The video demonstrates how to find the Laplace transform of t times j naught of a t by utilizing the multiplication by t property.

• The change of scale property is used to obtain the value of j naught of a t from j naught of t.

• The Laplace transform of j naught of a t is found to be 1 upon square root of a square plus s square.

• The multiplication by t property is then used to find the Laplace transform of t times j naught of a t, resulting in -s divided by (s square plus a square) raised to 3/2.