Lecture Notes: Finding Particular Integral in Difference Equations

Introduction

• Dr. Gendra Purohit presents a lecture on finding the particular integral in difference equations.
• Useful for students preparing for competitive exams in engineering mathematics.

Overview of Difference Equations

• The general form of the solution is CF (Complementary Function) plus PI (Particular Integral).
• CF was explained in the previous video.
• Aim: To solve for PI in numerical analysis.

Steps to Find the Particular Integral

1. Convert to Auxiliary Equation

   • The auxiliary equation is denoted as FE.
   • Similar to differential equations, manipulate the equation to find PI.

2. Functions of the Form a^n

   • For functions of type a^n, the auxiliary equation is utilized.

   • If e is substituted with a, ensure a is not equal to zero.

   • If a becomes zero, use the trick:

     • If denominator becomes zero, use e - a^m * f(e).
     • Differentiate m times to resolve zero denominators.

   • Example: For the equation Y(n+2) - 4Y(n+1) - 4YN = 2^n, find:

     • CF: C1 + C2n * 2^n
     • PI: derived from substituting e with 2 in the equation.

3. Functions involving Cosine or Sine

   • The approach is similar to differential equations.
   • For example: Y(n+2) - 7Y(n+1) + 12YN = cos(n).
   • Calculate CF first, then find the PI.

4. Functions of n^p Type

   • Convert to factorial notation first.
   • Use binomial expansion if necessary.
   • Example: Convert n^2 to factorial notation to derive PI.

Finding the Complementary Function (CF)

• Solve the auxiliary equation (e.g., e^2 - 7e + 12 = 0) to find roots.
• Use roots to determine the CF form:
  • Complex roots yield C1 * cos(n * theta) + C2 * sin(n * theta).

Finding the Particular Integral (PI)

• Substitute into the equation to find PI based on the CF.
• Use previously established rules for functions (e.g., a^n, cos, sin, n^p).

Conclusion

• The final solution is YN = CF + PI.
• Example problems demonstrate the process clearly.
• Students are encouraged to practice and refer to prior videos for complex topics.

Additional Resources

• Students preparing for exams like IIT Jam or CSIR NET can find full playlists available on the channel.
• Engage with the content by commenting on the lectures.