Math Problem Statement
please solve problem number 6 which great, insightful and step-by-step explanation!
Solution
The problem is asking to find the inverse Laplace transform of the function:
Let's solve this step-by-step.
Step 1: Understand the given function
The function can be written as:
We need to find the inverse Laplace transform of , i.e., we want to find such that:
Step 2: Decompose the function using Partial Fraction Decomposition
To make the inverse Laplace transform easier, we'll use partial fraction decomposition on .
Let’s express as:
To find , , and , multiply both sides by :
This expands to:
Now, equate the coefficients of , , and the constant term from both sides of the equation:
- Coefficient of :
- Coefficient of :
- Constant term:
Step 3: Solve for , , and
From and , we find:
So the partial fraction decomposition is:
Step 4: Find the inverse Laplace transforms
Now, take the inverse Laplace transform of each term separately.
- Inverse Laplace transform of :
- Inverse Laplace transform of :
Step 5: Combine the results
Combine the results from the inverse Laplace transforms:
Final Answer:
The inverse Laplace transform of is:
This function represents the solution in the time domain.
Would you like to see more details or have any questions?
Here are some related questions you might find helpful:
- How can the convolution theorem be applied to find the inverse Laplace transform of products like ?
- What are some common Laplace transforms and their corresponding time-domain functions?
- How does the inverse Laplace transform change if additional terms like are involved?
- How can partial fraction decomposition be used in solving differential equations with Laplace transforms?
- What are the applications of inverse Laplace transforms in solving real-world problems?
Tip: When working with Laplace transforms, keep a table of common transforms handy to quickly identify and apply the inverse transform.
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Inverse Laplace Transform
Partial Fraction Decomposition
Formulas
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Theorems
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Suitable Grade Level
Advanced
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