Problem 6 Based on Multiplication By 't' Property - Laplace Transform - Engineering Mathematics 3

TL;DR

This video explains how to apply the shifting and multiplication by t properties in Laplace transform, specifically for the error function of root t.

Transcript

hello friends so after solving many problems of multiplication by t property now we are going to solve or we are going to apply that property on error function of root t so let's see how to apply the property so here we have to find out the laplace of t into e to the power 3t and a function of root t now guys for this you have to go back because yo... Read More

Key Insights

• ❓ The shifting property is a useful technique to find Laplace transforms when the function can be considered as f(t).
• 💁 The multiplication by t property is applicable when the function is in the form of t * f(t).
• 🫚 The Laplace transform of the error function of root t can be derived using the multiplication by t property.
• 🍃 The final Laplace transform result involves simplifying the numerator and leaving the denominator as it is.
• 💼 It is recommended to solve the problem using the second method (shifting property) in this case.
• 🫚 The Laplace transform for t * error function of root t is obtained by taking the derivative of the Laplace transform of the error function of root t.

Questions & Answers

Q: How can the shifting property be used to find the Laplace transform in this case?

The shifting property involves treating the given function as f(t) and finding its Laplace transform, phi(s). Then, substitute s with s-3 in phi(s) to get the final answer.

Q: Can you explain the derivation of the Laplace transform for the error function of root t?

By using the multiplication by t property, the Laplace transform of the error function of root t is derived as 1/(s(sqrt(s)+1)). This can be found by considering the function as t * error function of root t and applying the multiplication by t property formula.

Q: What is the formula for the Laplace transform of t * error function of root t?

The Laplace transform of t * error function of root t is given by the formula -d/ds(1/sqrt(s+1)).

Q: What is the final result of the Laplace transform for t * e^(3t) * error function of root t?

The final result is (3s-7)/[(2(s-3))^2(sqrt(s-2))^3/2]. This is obtained by using the shifting property and the multiplication by t property.

Summary & Key Takeaways

• The video demonstrates how to find the Laplace transform of t * e^(3t) * error function of root t using the shifting property and multiplication by t property.

• The shifting property involves considering the function as f(t), finding its Laplace transform (phi(s)), and substituting phi(s) in the final answer as phi(s-3).

• By using the multiplication by t property, the Laplace transform of the error function of root t is derived as 1/(s(sqrt(s)+1)).

• The final result is the Laplace transform of t * e^(3t) * error function of root t = (3s-7)/[(2(s-3))^2(sqrt(s-2)^3/2)].