Finite Difference Operator | Numerical Analysis| UPSC ISS 2017 Paper-1 | Problem-43 | RitwikMath

Learn an important operator-based problem from **Finite Differences** and **Numerical Analysis** in this quick and exam-oriented solution.

We evaluate:

[
\frac{\Delta^2}{E}x^2
]

using the shift operator and finite difference identities.

📘 Key Concepts Used:
✔ Shift Operator:
[
Ef(x)=f(x+h)
]

✔ Difference Operator:
[
\Delta = E-1
]

Thus,
[
\Delta^2=(E-1)^2=E^2-2E+1
]

Therefore,
[
\frac{\Delta^2}{E}=E-2+E^{-1}
]

Applying this operator to (x^2):

[
(E-2+E^{-1})x^2
]

gives:

[
(x+h)^2-2x^2+(x-h)^2
]

which simplifies to:

[
2h^2
]

✅ Final Answer:
[
2h^2
]

✔ Option B is correct.

🎯 Useful for:
• UPSC Statistics Optional
• ISS Exam Preparation
• Numerical Analysis
• Engineering Mathematics
• CSIR NET / GATE / IIT JAM
• Mathematics Competitive Exams

📌 Topics Covered:
✔ Finite Difference Operators
✔ Shift Operator (E)
✔ Forward Difference Operator (\Delta)
✔ Operator Algebra
✔ Numerical Analysis MCQs
✔ Exam-Oriented Tricks

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