Solving ODE with Frobenius Form Solution

  • Thread starter Juggler123
  • Start date
  • Tags
    Frobenius
In summary, the conversation is about finding a solution to an ODE using a Frobenius form solution. The speaker has worked through the question and has ended up with a recurrence of the form a(k) = -2/(2k-1) a(k-1). They are trying to find a general recurrence in terms of a(0), but are having difficulty. The solution is expected to have a (2n-1)! term on the bottom of the fraction and a (-2)^(n) term on top. The speaker also mentions that there may be a point before the end where a(0) is established and it could potentially be 0. They clarify that they are not looking for anyone to write out the
  • #1
Juggler123
83
0
I'm finding a solution to and ODE using a Frobenius form solution, I have worked through the question and have ended up with a recurrence of the form;

a(k) = -2/(2k-1) a(k-1)

I'm trying to find a general reccurence in terms of a(0) but am finding it a bit difficult, I know it has to have a (2n-1)! term on the bottom of the fraction and a (-2)^(n) term on top but I'm a bit stuck as to where to go after this.

Sorry I haven't Latex-ed anything, I don't know how!

Any help would be fantastic, thanks.
 
Physics news on Phys.org
  • #2
good luck finding anyone who wants to actually write out frobenius method. If i remember correctly, there may be a point before the end where a(0) is established. and it could very well be 0. so heads up for that. otherwise, re-do it.
 
  • #3
I'm not looking for anyone to write out Frobenius method for me, I just need some help in simplifying the expression in terms of a(0), so that I can get a general expression for a solution. The problem dosen't have anything to do with Frobenius method at all.
 

FAQ: Solving ODE with Frobenius Form Solution

What is the Frobenius form solution?

The Frobenius form solution is a method for solving ordinary differential equations (ODEs) that cannot be solved using traditional methods, such as separation of variables or substitution. It involves transforming the ODE into a series of simpler equations, known as a power series, and then solving for the unknown function using algebraic techniques.

When is the Frobenius form solution used?

The Frobenius form solution is typically used when the ODE involves singular points, such as poles or branch points, and cannot be solved using other methods. It is also useful for solving higher-order ODEs with non-constant coefficients.

What are the steps involved in using the Frobenius form solution to solve an ODE?

The steps involved in using the Frobenius form solution are as follows:

  1. Transform the ODE into a power series by substituting the unknown function with a series expansion.
  2. Differentiate the series expansion and plug it into the ODE to obtain a recurrence relation for the coefficients.
  3. Solve the recurrence relation to obtain expressions for the coefficients in terms of the initial coefficients.
  4. Use the solutions for the coefficients to construct the general solution for the ODE.

What are the advantages of using the Frobenius form solution?

The Frobenius form solution allows for the solution of ODEs that cannot be solved using traditional methods. It also provides a systematic approach for solving higher-order ODEs and allows for the determination of the behavior of the solution near singular points.

What are the limitations of the Frobenius form solution?

The Frobenius form solution can be time-consuming and tedious, as it involves a series of algebraic manipulations. It is also only applicable to a limited set of ODEs and may not always yield a closed-form solution. Additionally, the series may not converge for certain values of the independent variable, which can limit the applicability of the solution.

Similar threads

MHB How Do You Initialize Negative Terms in a Frobenius Method Recurrence Relation?
Replies
1
Views
1K
Frobenius method for fourth order linear ODE
Replies
2
Views
3K
I Constants at the end of the Frobenius method
Replies
1
Views
2K
Method of Frobenius with exponential coefficients
Replies
1
Views
3K
Finding a solution to this equation using Frobenius method
Replies
1
Views
4K
Solving Coupled First Order ODEs: Is There a Closed Form Solution?
Replies
28
Views
2K
I Integrability of the tautological 1-form
Replies
1
Views
2K
MHB Solving Regular Singularities with Frobenius Expansion: Series Solution Question
Replies
10
Views
3K
Solution verification of ODE using Frobenius' method
Replies
3
Views
879
Solving ODEs w/ Frobenius Method: Q on b, c as Funcs of x
Replies
3
Views
6K

Hot Threads

Recent Insights

Back
Top