Determine positive integers A, B, C, D, E and F such that 12x-13 D Ex-F 6x²-13-5 A= C- D= E Bx+C +

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Title: Solving Rational Equations with Partial Fraction Decomposition**

**Determine Positive Integers \( A, B, C, D, E, \) and \( F \) such that:**

\[
\frac{12x - 13}{6x^2 - 13x - 5} = \frac{A}{Bx + C} + \frac{D}{Ex - F}
\]

**Objective:**

Find the positive integer values for \( A, B, C, D, E, \) and \( F \) that satisfy the equation above.

**Solution Steps:**

1. **Factor the Denominator:** Start by factoring the denominator \( 6x^2 - 13x - 5 \) to find possible linear factors. These factors will help form the denominators in the partial fractions on the right side of the equation.

2. **Decompose into Partial Fractions:** Once the denominator is factored, express the left-hand side as a sum of two separate fractions, each with linear denominators as shown.

3. **Determine Coefficients:** Equalize the numerators for both sides and solve for the unknowns \( A, B, C, D, E, \) and \( F \) by matching coefficients or by plugging in convenient values of \( x \) to generate a system of linear equations.

**Input Fields:**

- \( A = \) 
- \( B = \)
- \( C = \)
- \( D = \)
- \( E = \)
- \( F = \)

This method of decomposition will allow for the simplification and easier integration of rational expressions in calculus, or for solving other algebraic problems involving polynomial divisions.

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Transcribed Image Text:**Title: Solving Rational Equations with Partial Fraction Decomposition** **Determine Positive Integers \( A, B, C, D, E, \) and \( F \) such that:** \[ \frac{12x - 13}{6x^2 - 13x - 5} = \frac{A}{Bx + C} + \frac{D}{Ex - F} \] **Objective:** Find the positive integer values for \( A, B, C, D, E, \) and \( F \) that satisfy the equation above. **Solution Steps:** 1. **Factor the Denominator:** Start by factoring the denominator \( 6x^2 - 13x - 5 \) to find possible linear factors. These factors will help form the denominators in the partial fractions on the right side of the equation. 2. **Decompose into Partial Fractions:** Once the denominator is factored, express the left-hand side as a sum of two separate fractions, each with linear denominators as shown. 3. **Determine Coefficients:** Equalize the numerators for both sides and solve for the unknowns \( A, B, C, D, E, \) and \( F \) by matching coefficients or by plugging in convenient values of \( x \) to generate a system of linear equations. **Input Fields:** - \( A = \) - \( B = \) - \( C = \) - \( D = \) - \( E = \) - \( F = \) This method of decomposition will allow for the simplification and easier integration of rational expressions in calculus, or for solving other algebraic problems involving polynomial divisions. --- **Graph/Diagram Explanation:** There are no graphs or diagrams present in this image.
Expert Solution
Step 1

Solution: - The given equation is 12x-136x2-13x-5=ABx+C + DEx-F

then , we have to find the values of A,B,C,D,E,F ?

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