Math Problem Statement
L(6/(s^2-1)(s+2))
Solution
To find the Laplace transform , we'll first simplify the expression and then apply partial fraction decomposition.
Step 1: Factor the denominator
The expression is:
Factor :
So the expression becomes:
Step 2: Partial fraction decomposition
We express the function as a sum of simpler fractions:
Multiplying both sides by the denominator gives:
Step 3: Solve for A, B, and C
To find , , and , we substitute convenient values of .
- Let :
- Let :
- Let :
Thus, the decomposition is:
Step 4: Apply the inverse Laplace transform
Now, we apply the inverse Laplace transform to each term:
Final Answer:
Combining these, we get the inverse Laplace transform:
Would you like any further details or have any questions?
Here are 5 related questions you might find interesting:
- How do you perform partial fraction decomposition with repeated factors?
- What is the Laplace transform of ?
- How does the initial value theorem apply to Laplace transforms?
- Can you explain the convolution theorem in the context of Laplace transforms?
- What are some applications of Laplace transforms in engineering?
Tip: When performing partial fraction decomposition, check for repeated roots and handle them appropriately to avoid errors.
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Partial Fraction Decomposition
Inverse Laplace Transform
Formulas
L{f(t)} = ∫₀^∞ e^(-st)f(t) dt
Partial fraction decomposition: 6/(s-1)(s+1)(s+2) = A/(s-1) + B/(s+1) + C/(s+2)
Inverse Laplace Transforms: L⁻¹{1/(s-a)} = e^(at)
Theorems
Laplace Transform Theorem
Partial Fraction Decomposition
Suitable Grade Level
Grades 11-12 / College
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