How to derive Legendre Polynomials?

[terp.asessed]

Homework Statement

Could someone explain how Legendre polynomials are derived, particularly first three ones? I was only given the table in the class, not steps to solving them...so I am curious.

Homework Equations

P₀(x) = 1
P₁(x) = x
P₂(x) = 1/2 (3x² - 1)

The Attempt at a Solution

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[ShayanJ]

One way is solving the Legendre equation($\frac{d}{dx} [(1-x^2) \frac{dP_n}{dx}]+n(n+1)P_n=0$) using the method of power series.(Substituting $y=\sum_{n=0}^\infty a_n x^n$ in the equation and finding a recursive relation for the coefficients.)

 

[terp.asessed]

Thanks! The equation was NEVER given in the class, so I was stuck with the table, but again thanks!