User:Egm6321.f12.team2.alexander

=R*2.1=

Given
The following two equations:

Find
Prove $$L_2(y_H^1) = L2(y_H^2) = 0$$ where $$L_2(y)$$ is the Legendre Polynomial with n = 1. That is $$L_2(y):= (1-x^2)y'' - 2xy' +2y = 0$$.

Solution
Considering first $$y_H^1(x)$$, Taking the first differential gives Thus, Taking the second differential gives Thus, Taking $$L_2(y_H^1)$$ gives, Thus $$L_2(y_H^1)$$ = 0. Now,Considering for $$y_H^2(x)$$, Taking the first differential gives Taking the second differential gives Taking $$L_2(y_H^1)$$ gives,