Question Bank MCQ • 2 Marks Section 1: Mathematical Physics Differential equations

QB_PH_006: Differential equations

Question Statement

The Legendre polynomial $P_3(x)$ is given by:
(A) \frac{1}{2}(5x^3 - 3x)
(B) \frac{1}{2}(3x^3 - 5x)
(C) \frac{1}{2}(5x^2 - 1)
(D) \frac{1}{8}(35x^4 - 30x^2 + 3)
Correct Answer
A
Verified Key

Step-by-Step KaTeX Derivation & Detailed Explanation

### Solution: Rodrigues' formula for $n=3$: $P_3(x) = \frac{1}{48} \frac{d^3}{dx^3}(x^2-1)^3 = \frac{1}{2}(5x^3 - 3x)$.
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