Frobeniuns Method/Generalized Power Series to DiffEQ solutions

In summary: So, you could write y=\sum_{n=0}^{\infty} c_n x^{n+s}+x^{s-1}ory=\sum_{n=0}^{\infty} c_n x^{n+s}+\sum_{n=0}^{\infty} c_n(1-x^{s-1}) ory=\sum_{n=0}^{\infty} c_n(n+s)+x^{s-1} ory=\sum_{n=0}^{\infty} c_n(n+s)+
  • #1
mateomy
307
0
(Working out of Boas chapter 12, section 11)

[tex]
3xy'' + (3x + 1)y' + y = 0
[/tex]

I'm asked to solve the differential equation using the method of Frobenius but I'm finding the way Boas introduces/explains/exemplifies the method to be incredibly confusing. So, I used some google-fu and was even more confused. Seems like everyone has a different plan of attack for these problems.

What I've done so far is to assume
[tex]
y = \sum_{n=0}^{\infty} c_n x^{n+s}
[/tex]

*I know in a normal expansion it's simpy [itex]x^n[/itex] but from my understanding we're to multiply the summation by another factor of [itex]x^s[/itex], or whatever variable we choose.

...doing like wise for the respective derivatives:
[tex]
y' = \sum_{n=0}^{\infty} c_n (n+s) x^{n+s-1}
[/tex]

[tex]
y'' = \sum_{n=0}^{\infty} c_n (n+s-1) (n+s) x^{n+s-2}
[/tex]

then we substitute these expansions into the respective places within the original equation.

Now this is where I'm getting REALLY REALLY confused. Boas says to make a table and then from there find the indicial equation. From various pdf's and youtube videos I'm getting different information. Can anyone please point me onto the right path?

Thanks.
 
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  • #2
You just need to equate powers of $x$ by changing dumby variables of the different sums.

I can do this, though I will get another infraction from the moderators, so I better not.
 
  • #3
mateomy said:
(Working out of Boas chapter 12, section 11)

[tex]
3xy'' + (3x + 1)y' + y = 0
[/tex]

I'm asked to solve the differential equation using the method of Frobenius but I'm finding the way Boas introduces/explains/exemplifies the method to be incredibly confusing. So, I used some google-fu and was even more confused. Seems like everyone has a different plan of attack for these problems.

What I've done so far is to assume
[tex]
y = \sum_{n=0}^{\infty} c_n x^{n+s}
[/tex]

*I know in a normal expansion it's simpy [itex]x^n[/itex] but from my understanding we're to multiply the summation by another factor of [itex]x^s[/itex], or whatever variable we choose.

...doing like wise for the respective derivatives:
[tex]
y' = \sum_{n=0}^{\infty} c_n (n+s) x^{n+s-1}
[/tex]
So that [itex](3x+1)y'= \sum_{n=0}^\infty 3c_n(n+s)x^{n+s}+ \sum_{n=0}^\infty c_n(n+s)x^{n+s- 1}[/itex]

[tex]
y'' = \sum_{n=0}^{\infty} c_n (n+s-1) (n+s) x^{n+s-2}
[/tex]
So that [itex]3xy''= \sum_{n=0}^\infty 3c_n(n+s-1)(n+s)x^{n+s-1}[/itex].

then we substitute these expansions into the respective places within the original equation.

Now this is where I'm getting REALLY REALLY confused. Boas says to make a table and then from there find the indicial equation. From various pdf's and youtube videos I'm getting different information. Can anyone please point me onto the right path?

Thanks.
Putting those into your equations and combinging like powers will give sums in [itex]x^{n+s}[/itex] and [itex]x^{n+s-1}[/itex]. The crucial point is that we could choose "s" to be anything- we choose it so that the lowest power of x is [itex]x^0= 1[/itex]. That is, we choose s to be such that this infinite series contains a constant term. The lowest power of x in each sum is with n=0 so that the lowest power term is [tex]x^{s-1}[/itex]. You want to determine s so that coefficient is not 0. That is the "indicial equation".
 

1. What is the Frobenius method and how is it used to solve differential equations?

The Frobenius method is a powerful technique for finding solutions to differential equations that cannot be solved using standard methods. It involves assuming a power series solution and then using a recursive algorithm to determine the coefficients of the series. This method is particularly useful for finding solutions near singular points of the differential equation.

2. What are the advantages of using the Frobenius method over other methods for solving differential equations?

The Frobenius method allows for finding solutions to differential equations that cannot be solved using other methods. It is also more efficient and accurate for finding solutions near singular points. Additionally, the power series solution obtained using this method can be used to approximate solutions for a wider range of initial conditions.

3. Are there any limitations to the Frobenius method?

Yes, the Frobenius method is limited to solving linear differential equations with polynomial coefficients. It is also not applicable to all types of singular points, such as essential singular points. In some cases, the series solution obtained using this method may not converge or may only converge for a limited range of initial conditions.

4. Can the Frobenius method be extended to solve nonlinear differential equations?

Yes, there are some techniques for extending the Frobenius method to certain types of nonlinear differential equations. However, these methods are more complex and may not always yield a closed form solution. In general, the Frobenius method is most effective for solving linear differential equations.

5. How is the Frobenius method related to the generalized power series method?

The generalized power series method is a generalization of the Frobenius method that allows for finding solutions using more general types of power series. This method is useful for solving differential equations with non-polynomial coefficients or for finding solutions near more complicated types of singular points. The Frobenius method is a special case of the generalized power series method.

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