Math Problem Statement
Inverse Laplace in partial fraction
Solution
It looks like you've uploaded an image of a mathematical expression that you want to solve using the inverse Laplace transform in partial fractions. Let's first break down the expression in the image:
The expression shown is:
Steps to solve using Partial Fraction Decomposition:
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Express the function as a sum of simpler fractions: where and are constants to be determined.
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Multiply both sides by to get rid of the denominator: Simplify and group like terms:
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Set up a system of equations by equating coefficients:
- Coefficient of :
- Constant term:
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Solve the system of equations:
- From , we have .
- Substitute into the second equation: , which simplifies to , so , hence .
- Since , .
So, the partial fraction decomposition is:
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Apply the inverse Laplace transform:
- The inverse Laplace transform of is .
- Therefore:
Final Answer:
Would you like any further details or have any questions on this solution?
Here are some related questions you might consider:
- How does the Laplace transform of compare to this case?
- What happens if the denominator has repeated roots?
- How can partial fraction decomposition be applied to higher-degree polynomials?
- What is the significance of each term in the inverse Laplace transform?
- How would the procedure change if had a complex root?
Tip: When solving inverse Laplace problems, always make sure your decomposition is correct by re-multiplying to check your work before applying the inverse transform.
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Math Problem Analysis
Mathematical Concepts
Inverse Laplace Transform
Partial Fraction Decomposition
Formulas
Inverse Laplace of 1/(s-a) is e^{at}
Partial Fraction: 1/(s-2)(s+3) = A/(s-2) + B/(s+3)
Theorems
Laplace Transform Theorem
Partial Fraction Theorem
Suitable Grade Level
Undergraduate
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