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schamp
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Homework Statement
Show that the differential equation:
sin(theta)y'' + cos(theta)y' + n(n+1)(sin(theta))y = 0
can be transformed into Legendre's equation by means of the substitution x = cos(theta).
Homework Equations
Legendre's Equation:
(1 - x^2)y'' - 2xy' + n(n+1)y = 0
The Attempt at a Solution
Using the Chain Rule for the first derivative:
dy/d(theta) = (dy/dx)(dx/d(theta))
x = cos(theta)
dx/d(theta) = -sin(theta)
dy/d(theta) = -sin(theta)dy/dx
Using the Chain Rule for the second derivative:
d2y/d(theta)2 = (d2y/dx2)(d2x/d(theta2))
x = cos(theta)
d2x/d(theta)2 = -cos(theta)
d2y/d(theta)2 = d2y/dx2(-cos(theta))
After substitution I get:
-(sin(theta)*cos(theta))d2y/dx2 + -(sin(theta)*cos(theta))dy/dx + n(n+1)sin(theta)y = 0
I've played around with this and tried various trigonometric identities to get it into a form that can be translated as the Legendre's Equation, but I still can't quite get it in the form of:
(1-cos2(theta))d2y/dx2 - 2cos(theta)dy/dx + n(n+1)y = 0.