91Ó°ÊÓ

An equation similar to Legendre’s equation is-

(1-x²) y"-2xy'+[l(l+1)-m²/1-x²] y=0 (∀ m²â‰¤ l²)

The solution to this equation is of the form

P_l^m (x)=(1-x²)^m/2 d^m/dx^m (P_l(x))

These solutions are called Associated Legendre Functions.

The equation is given as,

1/sinθ d/dθ (sinθ »å²â/»åθ) + [l (l+1) -m²/sin²Î¸] y=0 …… (1)

Using x=cosθ in equation (1) to derive the equation,

(1-x²) y"-2xy+[l (l+1)-m²/1-x²] y=0

Putting x=cosθ , then:

1-x²= 1-cos²Î¸

=sin²Î¸

And,

dx/dθ = -sinθ

»å²â/»å³æ=»å²â/»åθ × »åθ/»å³æ

dy/dx= -1/sinθ × »å²â/»åθ

From the equation, obtain:

dy/dx = -1/sinθ »å²â/»åθ

»å²â/»åθ = -sinθ dy/dx

d²y/dx² = d/dx (dy/dx)

d²y/dx² = d/dθ (dy/dx) × »åθ/»å³æ

Simplify further as follows:

d²y/dx²= »åθ/»å³æ × d/dθ (-1/sinθ »å²â/»åθ)

d²y/dx²= -1/sinθ (-1/sin²Î¸ × d²y/dx² +³¦´Ç²õθ/²õ¾±²Ôθ × »å²â/»åθ)

d²y/dx²= 1/sin²Î¸ d²²â/»åθ² -cosθ /sinθ »å²â/»åθ

d²y/dx²= sin²Î¸ d²y/dx² +cosθ /sinθ »å²â/»åθ

As, sin²Î¸ =1-x² and »å²â/»åθ= -sinθ dy/dx

Hence,

d²y/dx² = (1-x²) d²y/dx² +cosθ /sinθ (-sinθ dy/dx)

d²y/dx² = (1-x²) d²y/dx² - x dy/dx

Now, rewriting equation (1) as following:

1/sinθ d/dθ (sinθ »å²â/»åθ) + [l (l+1) -m²/sin²Î¸] y=0

1/sinθ (sinθ d²²â/»åθ² + cosθ »å²â/»åθ) + [l (l+1)-m²/sin²Î¸] y=0

[d²²â/»åθ² -cos(-1/sinθ »å²â/»åθ)] + [l(l+1) -m²/sin²Î¸] y=0

Putting the known values in the above equation:

[(1-x²) d²y/dx² -x dy/dx-x (dy/dx)] + [l (l+1)-m²/sin²Î¸] y=0

(1-x²) y"-2xy+[l (l+1) - m²/1-x²] y=0

Hence, the required equation is (1-x²) y"-2xy+[l (l+1) - m²/1-x²] y=0.