Find the given inverse transform: L-1 1 s3+5s.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Title:** Finding the Inverse Laplace Transform
**Content:**
In this exercise, we are tasked with finding the inverse Laplace transform of the given function:
\[
L^{-1} \left\{ \frac{1}{s^3 + 5s} \right\}
\]
The expression inside the inverse transform involves a rational function, with the numerator being 1 and the denominator as a polynomial \(s^3 + 5s\).
To solve this, we will utilize partial fraction decomposition and properties of inverse Laplace transforms. The steps typically involve:
1. **Factor the denominator:** Refactor \(s^3 + 5s\) into simpler polynomials, if possible.
2. **Express as partial fractions:** Break down the expression into simpler, easily invertible fractions.
3. **Find the inverse Laplace transform:** Apply known inverse Laplace transforms to find the solution in the time domain.
By completing these steps, you will be able to determine the function in the time domain corresponding to the given Laplace transform.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89a69b77-6859-4c30-95b3-e6df6014c589%2F0ebf14f8-4221-451d-8400-0548a449dbd3%2Fvoytzw_processed.png&w=3840&q=75)
Transcribed Image Text:**Title:** Finding the Inverse Laplace Transform
**Content:**
In this exercise, we are tasked with finding the inverse Laplace transform of the given function:
\[
L^{-1} \left\{ \frac{1}{s^3 + 5s} \right\}
\]
The expression inside the inverse transform involves a rational function, with the numerator being 1 and the denominator as a polynomial \(s^3 + 5s\).
To solve this, we will utilize partial fraction decomposition and properties of inverse Laplace transforms. The steps typically involve:
1. **Factor the denominator:** Refactor \(s^3 + 5s\) into simpler polynomials, if possible.
2. **Express as partial fractions:** Break down the expression into simpler, easily invertible fractions.
3. **Find the inverse Laplace transform:** Apply known inverse Laplace transforms to find the solution in the time domain.
By completing these steps, you will be able to determine the function in the time domain corresponding to the given Laplace transform.
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