Solve Legendre Polynomial using Method of Frobenius

[feoxx]

Not sure how this can be done. can anyone help?

 

[craig_241]

Have you made an attempt?

If you don't know what the method of Frobenius is, then you should look it up. It usually comes under power series solutions of ODEs.

 

[feoxx]

I tried, i know that y''P(x)y'+Q(x)y = 0.
P(x) = (1-x)^2
Q(x) = (n^2+n)

but I am not sure what to do in terms of legendre polynomial.

 

[SteamKing]

The Legendre polynomials are the solutions to this type of ODE. For the moment, forget Legendre Polynomials.

What you must do is use the method of Frobenius to solve the given ODE. If you do it correctly, your solution should come out equal to the series expression given. Note: these solutions are series with a finite number of terms.

 

[blue_raver22]

The Method of Frobenius is a powerful mathematical tool that can be used to solve Legendre Polynomials. This method involves expressing the polynomial as a power series and then using a recursive formula to find the coefficients of the series.

To begin, we can express the Legendre Polynomial as:

P(x) = a0 + a1x + a2x^2 + a3x^3 + ...

We then substitute this into the differential equation for the Legendre Polynomial, which is:

(1-x^2)y'' - 2xy' + n(n+1)y = 0

Next, we differentiate the polynomial and plug it into the equation. This will give us a recursive formula for the coefficients a0, a1, a2, etc. Using this formula, we can solve for each coefficient and thus find the full power series representation of the Legendre Polynomial.

It is important to note that the value of n in the differential equation corresponds to the degree of the polynomial. So for example, if we are solving for the Legendre Polynomial of degree 3, we would use n=3 in the differential equation.

The Method of Frobenius may seem complex, but with careful application and a good understanding of power series, it can be used to successfully solve Legendre Polynomials. I hope this helps to clarify the process.