Why is answer A+B=1 and 2A-3B=-2 ?
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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Why is answer A+B=1 and 2A-3B=-2 ?
![The image shows a handwritten mathematical solution for the partial fraction decomposition of a rational function.
---
**Problem:**
Given the rational function:
\[ Y(s) = \frac{s - 2}{s^2 - s - 6} \]
Factor the denominator:
\[ s^2 - s - 6 = (s - 3)(s + 2) \]
The function can be decomposed into partial fractions as follows:
\[ \frac{s - 2}{(s - 3)(s + 2)} = \frac{A}{s - 3} + \frac{B}{s + 2} \]
**Steps to solve:**
1. **Clear the denominators:**
\[ s - 2 = A(s + 2) + B(s - 3) \]
2. **Expand and simplify:**
\[ A(s + 2) + B(s - 3) = As + 2A + Bs - 3B \]
Combine like terms:
\[ (A + B)s + (2A - 3B) \]
3. **Compare coefficients:**
For the expression to equal \( s - 2 \), equate coefficients:
\[ A + B = 1 \]
\[ 2A - 3B = -2 \]
This system of equations can be solved to find \( A \) and \( B \).
There are no graphs or additional diagrams included in the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b9f9fa7-cab4-4dbd-8980-f3534e4105e4%2F932694c1-9d2f-4076-8bee-82ed7375136f%2Frc9znjk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image shows a handwritten mathematical solution for the partial fraction decomposition of a rational function.
---
**Problem:**
Given the rational function:
\[ Y(s) = \frac{s - 2}{s^2 - s - 6} \]
Factor the denominator:
\[ s^2 - s - 6 = (s - 3)(s + 2) \]
The function can be decomposed into partial fractions as follows:
\[ \frac{s - 2}{(s - 3)(s + 2)} = \frac{A}{s - 3} + \frac{B}{s + 2} \]
**Steps to solve:**
1. **Clear the denominators:**
\[ s - 2 = A(s + 2) + B(s - 3) \]
2. **Expand and simplify:**
\[ A(s + 2) + B(s - 3) = As + 2A + Bs - 3B \]
Combine like terms:
\[ (A + B)s + (2A - 3B) \]
3. **Compare coefficients:**
For the expression to equal \( s - 2 \), equate coefficients:
\[ A + B = 1 \]
\[ 2A - 3B = -2 \]
This system of equations can be solved to find \( A \) and \( B \).
There are no graphs or additional diagrams included in the image.
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