Laplace Transform - First Shifting Theorem with Example | By GP Sir
Summary
TLDRIn this educational video, the instructor introduces the 'First Shifting Theorem' in the context of Laplace Transforms, focusing on its application for both direct and inverse transformations. The video simplifies the concept with straightforward examples, emphasizing the importance of understanding the theorem's implications rather than its proof. The instructor demonstrates how to apply the theorem to find the Laplace Transform of given functions and then invert them, using additional examples for clarity. The lecture aims to ensure comprehension and encourages students to engage with any lingering questions in the comment section. The video concludes with a call to action for viewers to share and subscribe to the channel.
Takeaways
- 📚 The video discusses the 'First Shifting Theorem' in the context of Laplace Transforms.
- 🔍 The theorem is used to find the Laplace Transform and its inverse.
- 📝 The presenter provides simple statements of the theorem without focusing on the proof.
- 📈 Three examples are given to demonstrate how to find the Laplace Transform using the theorem.
- 🔄 The 'First Shifting Theorem' is also applicable for inverse Laplace Transforms.
- 📑 The video includes examples to illustrate finding the inverse Laplace Transform.
- 🤔 The presenter assures that the concept is simple and that proofs will be taught when necessary.
- 👨🏫 The lecture aims to ensure understanding of the first shifting theorem for both Laplace and inverse Laplace Transforms.
- 💬 Students are encouraged to ask questions in the comment section if they have doubts.
- 📢 The presenter invites viewers to share and subscribe to the channel for more educational content.
Q & A
What is the 'First Shifting Theorem' in the context of Laplace Transform?
-The 'First Shifting Theorem' is a principle in Laplace Transform that allows for the simplification of finding the Laplace Transform of functions that are shifted in time. It is a simple concept that is crucial for solving problems involving time delays or advanced functions.
How does the 'First Shifting Theorem' simplify the process of finding Laplace Transforms?
-The 'First Shifting Theorem' simplifies the process by providing a direct formula to calculate the Laplace Transform of a function that is time-shifted. This avoids the need for complex integrations and makes it easier to handle functions with time delays.
What are the implications of the 'First Shifting Theorem' that students need to remember?
-Students need to remember that the 'First Shifting Theorem' allows them to easily find the Laplace Transform of functions that are shifted in time by a certain amount. The theorem provides a straightforward way to account for this time shift in the transform.
Is it necessary to understand the proof of the 'First Shifting Theorem' to use it effectively?
-While understanding the proof can be beneficial, it is not necessary for practical use. The focus should be on the application of the theorem to solve problems, and the proof can be studied when needed for a deeper understanding.
Can you provide an example of how the 'First Shifting Theorem' is applied to find the Laplace Transform?
-Sure, if you have a function f(t-a)u(t-a) where u(t) is the unit step function, the Laplace Transform using the 'First Shifting Theorem' would be e^{-as}F(s) where F(s) is the Laplace Transform of f(t).
What is the inverse process of the 'First Shifting Theorem'?
-The inverse process of the 'First Shifting Theorem' is used to find the inverse Laplace Transform of a function. It helps in determining the original time function when given its Laplace Transform, especially when the function has been shifted in the s-domain.
How does the 'First Shifting Theorem' assist in finding the inverse Laplace Transform?
-The 'First Shifting Theorem' assists in finding the inverse Laplace Transform by providing a method to shift the function back in time. It allows for the reconstruction of the original time-domain function from its Laplace Transform representation.
Can you give an example of using the 'First Shifting Theorem' for inverse Laplace Transform?
-If the Laplace Transform of a function is given as e^{-as}F(s), using the 'First Shifting Theorem' for inverse Laplace Transform, the original function can be found as f(t-a)u(t-a) where F(s) is the Laplace Transform of f(t).
What are the key takeaways from the lecture on the 'First Shifting Theorem'?
-The key takeaways include understanding what the 'First Shifting Theorem' is, how to use it to find Laplace Transforms of time-shifted functions, and how to apply it to find the inverse Laplace Transform. The lecture also emphasized the importance of knowing the implications of the theorem without necessarily focusing on the proof.
How can students ensure they understand the 'First Shifting Theorem' and its applications?
-Students can ensure their understanding by practicing with various examples, reviewing the theorem's implications, and asking questions in the comment section if they have doubts. Engaging with the material through practice and discussion is key to solidifying the concepts.
Outlines
📚 Introduction to First Shifting Theorem
The speaker begins by addressing the students and introducing the topic of the video, which is the 'First Shifting Theorem' in Laplace Transform. They mention that they will cover how to use this theorem for both direct and inverse Laplace Transforms. The speaker reassures the audience that although there is a proof for the theorem, it is not necessary to focus on it for this lesson. Instead, the emphasis is on understanding the implications of the theorem. The speaker provides examples to demonstrate how to find the Laplace Transform using the First Shifting Theorem and promises to teach the proof when needed.
🔍 Applying First Shifting Theorem in Practice
In this paragraph, the speaker delves into practical applications of the First Shifting Theorem by providing examples. They illustrate how to solve problems using the theorem and then move on to teach how to find the inverse Laplace Transform using the inverse shifting theorem. The speaker takes the audience through multiple examples to solidify their understanding of applying the First Shifting Theorem to both direct and inverse Laplace Transforms.
📢 Conclusion and Call to Action
The speaker concludes the lecture by summarizing the key points covered in the video. They have taught the concept of the First Shifting Theorem, how to use it to find the Laplace Transform, and subsequently, how to find the inverse Laplace Transform. The speaker expresses hope that the students have understood the material and encourages them to ask questions in the comment section if they have any doubts. They also invite the audience to share the video, subscribe to the channel, and engage with the content.
Mindmap
Keywords
💡Laplace Transform
💡First Shifting Theorem
💡Inverse Laplace Transform
💡Time Domain
💡s-plane
💡Linear Time-Invariant Systems
💡Differential Equations
💡Proof
💡Comment Section
💡Subscription
Highlights
Introduction to the use of 'First Shifting theorem' in Laplace Transform.
Explanation of how to use 'First Shifting theorem' in inverse Laplace Transform.
Guidance on finding the Laplace transform by using the first shifting theorem.
Emphasis on the simplicity of the 'First Shifting theorem' concept.
Clarification that the proof of the theorem is not a focus, but its implications are.
Illustration of the theorem's application through two examples.
Introduction of the concept of inverse first shifting theorem.
Demonstration of how to find the inverse Laplace of given functions.
Explanation of the 'First Shifting theorem' for inverse Laplace Transform.
Presentation of three examples to find Laplace using the first shifting theorem.
Instruction on finding the inverse Laplace using the inverse shifting theorem.
Discussion on solving questions using the first shifting theorem.
Presentation of additional examples to solidify understanding.
Summary of the lecture's content on the first shifting theorem.
Encouragement for students to ask questions in the comment section if they have doubts.
Request for viewers to share and subscribe to the channel for more content.
Transcripts
Hello students, today I'm here with another video
use of 'First Shifting theorem' in Laplace Transform,
how to use it in inverse as well
and how to find Laplace transform by first shifting theorem
and inverse first shifting theorem
Previously, I taught you the concept and formulas of Laplace theorem
so today we will talk about 'First Shifting theorem'
What is 'first shifting theorem'? Let me tell you its simple statement
This is a very simple concept that you need to remember
Although it has proof, but we don't need to focus on that
all we need to know is its implication
I'll teach you the proof whenever it will be needed
So have a look here
If you are asked to find the Laplace of this
I am taking these two examples
Let me take one more example
Now I'll teach you
how to find the inverse Laplace of the same questions
For this, we will have to learn
'First Shifting theorem' for inverse Laplace Transform
Above, I took three examples
on how to find Laplace using first shifting theorem
Now I'll teach you to find its inverse Laplace using inverse shifting theorem
First Shifting Theorem for Inverse Laplace Transform
What I want to say is
For example, let's take a question
So this is how we solve such type of questions
I'll take one or two more examples then we will end this topic
Have a look at these two examples
So students, today I taught you
What is first shifting theorem, how to find Laplace transform using this
then we studied for the same questions
how to find the inverse Laplace using 'first Laplace theorem'
I hope you were able to understand the complete lecture
and you won't be having any doubt now
but still, if you have, then keep asking them in the comment section
If you like my videos, then please share them
do subscribe to my channel
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