Lecture Notes: Finding Particular Integral in Difference Equations

Introduction

• Dr. Gendra Purohit introduces the lecture on finding particular integrals in difference equations.
• Emphasis on the relevance of the topic for engineering mathematics and competitive exams.

Key Concepts

Linear Difference Equations

• The general solution to a difference equation is given by: CF + PI (Complementary Function + Particular Integral).
• Previous videos explain how to find CF. This lecture focuses on finding PI.

Auxiliary Equation

• Convert the difference equation into its auxiliary equation.
• Example: For a function P(X), use the formula P(X)/FD to find the particular integral.

Steps to Find Particular Integral

A^n Type Functions

1. Formulation: For a function of the form A^n, auxiliary equation is taken down:
   • YN = A^n / (E^s - 4E - 4) , where E is the shift operator.
2. Special Case (A = 0): If A becomes zero, use the formula:
   • E - A^m / M! where derivatives are employed.
3. Example: Equation: YN + 2 - 4YN + 1 - 4YN = 2^n
   • Resulting in YN = 2^n / (E^2 - 4E - 4)
   • Find CF, then find PI by substituting values.

Cosine and Sine Functions

1. Similar to differential equations, if functions are cos(kN) or sin(kN), apply same methods:
   • Example: YN + 2 - 7YN + 1 + 12YN = cos(n)
2. Calculation of CF: Use characteristic equation to find roots.
3. Calculation of PI: Substitute values and expand.

n^P Type Functions

1. Convert to factorial notation if function is n^P.
2. Use binomial expansion once shifted operator is converted to D.
3. Example: For YN + 2 + YN + 1 = n^2,
   • Convert n^2 into factorial notation to simplify calculations.

Product Functions (A^n * f(N))*

1. For products like A^n * f(N), take out A^n and deal with the remaining function:
   • Example: YN + 1 - 3YN = n^2 * 2^n.
2. Solve for CF and PI separately.

Examples and Detailed Solutions

• The lecture includes multiple detailed examples illustrating the above methods, including:
  • Finding CF and PI for equations involving powers, trigonometric functions, and product functions.
  • Comments on how to simplify and expand expressions for ease of computation.

Conclusion

• Dr. Purohit encourages students to practice these methods for better understanding and preparation for exams.
• A reminder to check previous videos for foundational concepts.

Additional Resources

• Playlist available for numerical analysis and competitive exam preparation.
• Encouragement to subscribe for more educational content.