Math Problem Statement
solve using inverse laplace
Solution
The equation in the image you provided represents a Laplace transform:
To solve for the inverse Laplace transform of this function, we will first attempt to factorize the denominator, and then we can use known inverse Laplace transform techniques.
Step 1: Factorizing the Denominator
We begin by factoring the quadratic expression :
Thus, the function becomes:
Step 2: Applying the Inverse Laplace Transform
The presence of in the numerator indicates a time shift property in the Laplace transform. Using the inverse Laplace transform properties and the known inverse transforms, we can start by looking at the function , which can be decomposed into partial fractions.
We express:
Multiplying through by , we get:
By solving for and , we find:
- Set :
- Set :
Thus, the partial fraction decomposition is:
Step 3: Inverse Laplace Transform
Now, using the inverse Laplace transform properties:
Therefore, we have:
Step 4: Time Shifting Property
The term in the numerator of the original Laplace transform indicates a time shift of 1 unit. Using the time-shifting property of the inverse Laplace transform:
where is the unit step function shifted to , and is the inverse Laplace transform of the function without the term.
Thus, the solution becomes:
This is the inverse Laplace transform of the given function.
Would you like further details or have any questions about this process?
Here are some follow-up questions for your exploration:
- How do you apply the inverse Laplace transform to a function with higher-order polynomials in the denominator?
- What is the significance of the time-shifting property in Laplace transforms?
- Can you apply partial fraction decomposition to other types of rational functions, and how does the complexity change?
- How would this process change if the Laplace transform had higher powers of ?
- What are other common Laplace transform pairs and their applications?
Tip: In inverse Laplace transforms, always look for possible simplifications like factoring denominators or recognizing time shifts to streamline your work.
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Math Problem Analysis
Mathematical Concepts
Inverse Laplace Transform
Partial Fractions
Time Shifting Property
Formulas
F(s) = e^(-s) / (s(s + 2))
Time-shifting property: L^(-1)[e^(-as)F(s)] = u(t-a)f(t-a)
Theorems
Time Shifting Theorem
Partial Fraction Decomposition
Suitable Grade Level
Undergraduate Level
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