Problem 3 Based on Inverse Laplace Transform using Shifting theorem - Engineering Mathematics 3

TL;DR

Learn how to find the inverse Laplace transform of a complex function using the shifting theorem.

Transcript

hello students so after solving two numericals on shifting theorem let's move to the next numerical which is again based on inverse laplace transform using shifting theorem so here i have changed the function of s and i've made it a difficult than the previous problem so let's see how to get the answer of that so here we have to find out l inverse ... Read More

Key Insights

• 💁 The shifting theorem is a powerful tool that simplifies the calculation of the inverse Laplace transform by converting the function into a more manageable form.
• ❓ Partial fraction decomposition is a crucial step in finding the inverse Laplace transform of functions with complex denominators.
• 🆘 Understanding the properties of quadratic equations helps in simplifying the function and finding the necessary adjustments for partial fraction decomposition.

Questions & Answers

Q: What is the purpose of using the shifting theorem in finding the inverse Laplace transform?

The shifting theorem allows us to convert the function into the form "phi(s + a)", which simplifies the calculation of the inverse Laplace transform.

Q: How does the instructor simplify the function before applying the shifting theorem?

The instructor uses the property of quadratic equations to transform the denominator into perfect square terms, making it easier to manipulate algebraically.

Q: What is the significance of performing partial fraction decomposition?

Partial fraction decomposition allows us to express the function as a sum of simpler fractions, making it easier to find the inverse Laplace transform using known integral formulas.

Q: How does the instructor find the values of 'a' and 'b' in the partial fraction decomposition?

The instructor uses a substitution method, replacing 's^2' with 'x' to convert the quadratic equations into linear equations. The values of 'a' and 'b' are then determined by equating coefficients.

Summary & Key Takeaways

• The content explains how to find the inverse Laplace transform of a function by using the shifting theorem and partial fraction decomposition.

• The instructor demonstrates the step-by-step process of converting the function into the form "phi(s + a)" and applying the shifting theorem.

• The video also provides a detailed explanation of how to perform partial fraction decomposition to simplify the function further.

• The final solution is obtained by finding the inverse Laplace transform of the simplified function.