4.3 Partial Fraction Decomposition

Partial fraction decomposition breaks complex rational functions into simpler fractions, making integration and differential equation solving easier. It's a powerful technique that simplifies complex mathematical problems by breaking them down into manageable parts.

This method is crucial for integrating rational functions and solving differential equations. By decomposing complex fractions, we can tackle problems that would otherwise be too difficult to solve directly, opening up new possibilities in calculus and engineering.

Partial Fraction Decomposition

Concept of partial fraction decomposition

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• Breaks down complex rational functions into simpler fractions called partial fractions
• Each partial fraction has a denominator that factors the original denominator
• Based on the idea that rational functions can be written as sums of partial fractions with linear or irreducible quadratic denominators
• Involves finding coefficients of partial fractions
  • Coefficients determined by solving linear equations or using heaviside cover-up method

Decomposition of rational functions

• Proper rational functions have numerators with lower degrees than denominators
  • Decomposition consists only of partial fractions sum
• Improper rational functions have numerators with degrees greater than or equal to denominators
  • Decomposition has polynomial part and proper rational part
  • Polynomial part obtained by long division of numerator by denominator
  • Proper rational part decomposed into partial fractions
• Decomposition steps:
  1. Factor denominator into distinct linear or irreducible quadratic factors
  2. Set up partial fraction decomposition with unknown coefficients for each denominator factor
  3. Multiply equation by original denominator to clear fractions
  4. Equate coefficients of like terms to form linear equations system
  5. Solve equations system to find unknown coefficients values

Integration using partial fractions

• Simplifies integration of rational functions
• Integration steps:
  1. Perform partial fraction decomposition on rational function
  2. Integrate each partial fraction separately
     • Linear denominators use natural logarithm: $\int \frac{1}{ax+b} dx = \frac{1}{a} \ln|ax+b| + C$
     • Irreducible quadratic denominators use inverse tangent: $\int \frac{1}{ax^2+bx+c} dx = \frac{1}{\sqrt{4ac-b^2}} \arctan(\frac{2ax+b}{\sqrt{4ac-b^2}}) + C$
  3. Add individual integral results for final answer

Differential equations with partial fractions

• Solves linear differential equations with constant coefficients
• Useful when right-hand side is rational function
• Solution steps:
  1. Take Laplace transform of differential equation sides
  2. Isolate transformed dependent variable on one equation side
  3. Perform partial fraction decomposition on resulting rational function
  4. Find inverse Laplace transform of each partial fraction
  5. Add inverse Laplace transform results for general differential equation solution
  6. Apply initial or boundary conditions for particular solution