Determine the partial fraction expansion for the rational function below. - 19s + 75 (s+5) (s² +9) - 19s +75 (s+5) (s² +9)
Determine the partial fraction expansion for the rational function below. - 19s + 75 (s+5) (s² +9) - 19s +75 (s+5) (s² +9)
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: Partial Fraction Expansion Guide**
**Topic: Partial Fraction Expansion for Rational Functions**
---
**Objective**: Determine the partial fraction expansion for the rational function provided below.
\[ \frac{-19s + 75}{(s + 5)(s^2 + 9)} \]
---
### Steps to Solve the Problem:
1. **Identify the Rational Function**:
The given rational function is:
\[ \frac{-19s + 75}{(s + 5)(s^2 + 9)} \]
2. **Set Up the Partial Fraction Expansion**:
To expand this fraction, express it as a sum of simpler fractions:
\[ \frac{-19s + 75}{(s + 5)(s^2 + 9)} = \frac{A}{s + 5} + \frac{Bs + C}{s^2 + 9} \]
Here, \(A\), \(B\), and \(C\) are constants to be determined.
3. **Combine the Fractions**:
The goal is to express the right-hand side with a common denominator:
\[ \frac{A(s^2 + 9) + (Bs + C)(s + 5)}{(s + 5)(s^2 + 9)} \]
4. **Equate the Numerators**:
Set the numerator on the left side equal to the numerator on the right:
\[ -19s + 75 = A(s^2 + 9) + (Bs + C)(s + 5) \]
---
**Pending Solution**:
We now need to solve for \(A\), \(B\), and \(C\) by expanding and equating coefficients of like terms on both sides of the equation.
\[ -19s + 75 = A(s^2 + 9) + Bs^2 + 5Bs + Cs + 5C \]
Combine like terms:
\[ -19s + 75 = As^2 + 9A + Bs^2 + 5Bs + Cs + 5C \]
Group the coefficients of \(s^2\), \(s\), and the constants:
\[ -19s + 75 = (A + B)s^2 + (5B + C)s + (9A + 5C) \]
Compare coefficients for \(s^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1d700aa-072a-4ff7-be1d-cdca0f10aa9d%2F7e2c8694-b30c-4602-9c50-0903c7c08bde%2F7d85f0c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Partial Fraction Expansion Guide**
**Topic: Partial Fraction Expansion for Rational Functions**
---
**Objective**: Determine the partial fraction expansion for the rational function provided below.
\[ \frac{-19s + 75}{(s + 5)(s^2 + 9)} \]
---
### Steps to Solve the Problem:
1. **Identify the Rational Function**:
The given rational function is:
\[ \frac{-19s + 75}{(s + 5)(s^2 + 9)} \]
2. **Set Up the Partial Fraction Expansion**:
To expand this fraction, express it as a sum of simpler fractions:
\[ \frac{-19s + 75}{(s + 5)(s^2 + 9)} = \frac{A}{s + 5} + \frac{Bs + C}{s^2 + 9} \]
Here, \(A\), \(B\), and \(C\) are constants to be determined.
3. **Combine the Fractions**:
The goal is to express the right-hand side with a common denominator:
\[ \frac{A(s^2 + 9) + (Bs + C)(s + 5)}{(s + 5)(s^2 + 9)} \]
4. **Equate the Numerators**:
Set the numerator on the left side equal to the numerator on the right:
\[ -19s + 75 = A(s^2 + 9) + (Bs + C)(s + 5) \]
---
**Pending Solution**:
We now need to solve for \(A\), \(B\), and \(C\) by expanding and equating coefficients of like terms on both sides of the equation.
\[ -19s + 75 = A(s^2 + 9) + Bs^2 + 5Bs + Cs + 5C \]
Combine like terms:
\[ -19s + 75 = As^2 + 9A + Bs^2 + 5Bs + Cs + 5C \]
Group the coefficients of \(s^2\), \(s\), and the constants:
\[ -19s + 75 = (A + B)s^2 + (5B + C)s + (9A + 5C) \]
Compare coefficients for \(s^
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