Problem 1 on Change of Scale Property - Laplace Transform - Engineering Mathematics 3

TL;DR

The Laplace transform is used to evaluate the given integration, resulting in the value of 1/root 6.

Transcript

hello friends so here laplace of error function of root t is given as 1 upon s root of s plus 1 and we have to evaluate integration 0 to infinity e raised to minus 2 t error function of 2 root t dt now to get the value of this integration let's observe the integration so here we have the limits from 0 to infinity uh integration is with respect to t... Read More

Key Insights

• 🔨 The Laplace transform is a powerful tool for evaluating integrals involving functions.
• ⚖️ The change of scale property can simplify calculations when applying the Laplace transform.
• 🫚 The Laplace transform of the error function of 2 root t is derived using the Laplace transform definition and the change of scale property.

Questions & Answers

Q: How is the Laplace transform used to evaluate the integration?

The Laplace transform definition states that the Laplace transform of a function f(t) is equal to the integral of e^(-st) multiplied by f(t) over the interval from 0 to infinity. By applying this definition with the given function, we can find the Laplace transform of the error function of 2 root t.

Q: How is the change of scale property applied in this analysis?

The change of scale property states that the Laplace transform of f(a*t) is equal to (1/a) times the Laplace transform of f(t), where a is the coefficient of t. In this analysis, the coefficient of root t is 2, so we need to take 2 inside the square root to make it the coefficient of t.

Q: What is the Laplace transform of the error function of 2 root t?

The Laplace transform of the error function of 2 root t is found to be 1 upon s root of s plus 1, where s is the Laplace transform variable.

Q: What is the final value of the given integration?

After evaluating the Laplace transform and substituting s as 2, the value of the integration from 0 to infinity e^(-2t) error function of 2 root t dt is found to be 1 upon root 6.

Summary & Key Takeaways

• The given integration is evaluated using the Laplace transform.

• The Laplace transform definition is applied to the given function, resulting in the Laplace transform of the error function of 2 root t.

• The change of scale property is used to simplify the Laplace transform expression.

• The Laplace transform of the error function of 2 root t is found to be 1 upon root of 6.