Leading term of a power-series solution

[StephvsEinst]

Hey! I have a question: how can you find the power-series solution and its leading term of the following ODE around the point x=+-(1-b) (where b is near x):

$$-(1-x^2)\frac{\partial^2 f^m_l}{\partial x^2}+2x\frac{f^m_l}{\partial x}+\frac{m^2}{1-x^2}f^m_l=l(l+1)f^m_l ,$$

and m and l are constants.

EDIT: Let me ask a better question - how can you find the solution to the following ODE:

$$ sin(\theta)\frac{\partial}{\partial \theta}(sin(\theta)\frac{\partial f(\theta)}{\partial \theta})+[l(l+1)sin^2(\theta)-m^2)]f(\theta)=0 $$

Thank you.

 

[mfb]

I didn't solve it, but defining f(θ)=g(sin(θ)) makes the equation easier. Maybe the more general f(θ)=g(sin(θ)) + h(cos(θ)) is even better.

 

[StephvsEinst]

I already solved it: it isn't an homonegeous ODE, so to calculate its power-series solution you substitute x=cos(θ)=sqrt(1-(sin(θ))^2). The solution is:

$$\Theta(\theta)=A_l^mP_l^m(x)$$

where
$$P_l^m(x)\equiv (1-x^2)^{\vert m \vert/2}\left(\frac{d}{dx}\right)P_l(x)$$.
This is its associated Legendre function.

 

[StephvsEinst]

The solution is:

$$\Theta(\theta)=A_l^mP_l^m(x)$$

What I meant was that the solution is:
$$\Theta(\theta)=A_l^mP_l^m(cos\theta)$$

where

$$P_l^m(x)\equiv (1-x^2)^{\vert m \vert/2}\left(\frac{d}{dx}\right)P_l(x),$$

where

$$x=cos\theta.$$

 

[CellCoree]

Hello! Thank you for your question. The power-series method is a common technique used to solve ordinary differential equations (ODEs) around a given point. In this method, we assume that the solution can be expressed as a power series in the variable x, with unknown coefficients. The leading term of this series is the term with the highest power of x.

To find the power-series solution for the first ODE you provided, we can follow these steps:

1. Substitute the given power-series form into the ODE and equate the coefficients of each power of x to zero.

2. Solve the resulting system of equations to find the values of the coefficients.

3. Substitute these coefficients back into the power-series form to obtain the solution.

The leading term of this solution will be the term with the highest power of x.

Regarding your second question, the ODE you provided is known as the Legendre differential equation, which is commonly used in physics and engineering. The power-series method can also be used to find its solution. The leading term of this solution will depend on the values of the constants m and l.

I hope this helps answer your question. Let me know if you have any further doubts or need clarification. Good luck with your studies!