Answer to Question #333182 in Physics for Johannes Steven

Question

5.​ Obtain the value of P₃ (x) using Rodrigues' formula​​​​​       

P_n(x) = 1/(2ⁿ)n! x dⁿ/dxⁿ *(x²-1)ⁿ

Answer 1

$$P_3(x) = \frac{1}{(2^3)3!} \frac{d^3}{dx^3} (x^2-1)^3$$

$$= \frac{1}{48} \frac{d^2}{dx^2} (6x(x^2-1)^2)$$

$$= \frac{1}{8} \frac{d}{dx} ((x^2-1)^2+4x^2(x^2-1))$$

$$= \frac{1}{8} (4x(x^2-1)+8x(x^2-1)+8x^3)$$

$$= \frac{1}{2}(5x^3-3x)$$