Find the Laplace inverse L-¹. ¹ s²+6s+9 (s-1)(s-2)(s+4))
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Problem 1RQ
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![### Find the Laplace Inverse
Calculate the inverse Laplace transform denoted as \( \mathcal{L}^{-1} \) of the following function:
\[
\mathcal{L}^{-1} \left\{ \frac{s^2 + 6s + 9}{(s - 1)(s - 2)(s + 4)} \right\}
\]
#### Explanation:
In this problem, we are given a rational function in the s-domain and are required to find its inverse Laplace transform, which will yield a time-domain function \( f(t) \). The provided function is a fraction where the numerator is \( s^2 + 6s + 9 \) and the denominator is the product of three linear factors: \( (s-1)(s-2)(s+4) \).
You will likely need to use partial fraction decomposition to separate the complex fraction into simpler fractions that you can more easily invert using known inverse Laplace transforms. Here's a brief outline of the process:
1. **Decompose into Partial Fractions:**
Express the given fraction as a sum of simpler fractions.
\[
\frac{s^2 + 6s + 9}{(s-1)(s-2)(s+4)} = \frac{A}{s-1} + \frac{B}{s-2} + \frac{C}{s+4}
\]
2. **Solve for A, B, and C:**
Determine the constants \( A, B, \) and \( C \) by multiplying both sides by the denominator and equating coefficients.
3. **Apply the Inverse Laplace Transform:**
Use known inverse Laplace transforms on the simpler fractions.
By following these steps, you can find the required time-domain function. For detailed examples and more thorough explanations, consider reviewing materials on partial fraction decomposition and properties of the Laplace transform.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1d700aa-072a-4ff7-be1d-cdca0f10aa9d%2F0c87ca68-3314-4019-8445-6b5398487ad6%2Fpkxe1x_processed.png&w=3840&q=75)
Transcribed Image Text:### Find the Laplace Inverse
Calculate the inverse Laplace transform denoted as \( \mathcal{L}^{-1} \) of the following function:
\[
\mathcal{L}^{-1} \left\{ \frac{s^2 + 6s + 9}{(s - 1)(s - 2)(s + 4)} \right\}
\]
#### Explanation:
In this problem, we are given a rational function in the s-domain and are required to find its inverse Laplace transform, which will yield a time-domain function \( f(t) \). The provided function is a fraction where the numerator is \( s^2 + 6s + 9 \) and the denominator is the product of three linear factors: \( (s-1)(s-2)(s+4) \).
You will likely need to use partial fraction decomposition to separate the complex fraction into simpler fractions that you can more easily invert using known inverse Laplace transforms. Here's a brief outline of the process:
1. **Decompose into Partial Fractions:**
Express the given fraction as a sum of simpler fractions.
\[
\frac{s^2 + 6s + 9}{(s-1)(s-2)(s+4)} = \frac{A}{s-1} + \frac{B}{s-2} + \frac{C}{s+4}
\]
2. **Solve for A, B, and C:**
Determine the constants \( A, B, \) and \( C \) by multiplying both sides by the denominator and equating coefficients.
3. **Apply the Inverse Laplace Transform:**
Use known inverse Laplace transforms on the simpler fractions.
By following these steps, you can find the required time-domain function. For detailed examples and more thorough explanations, consider reviewing materials on partial fraction decomposition and properties of the Laplace transform.
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