How to solve a second order linear homeogeneous ODE with Frobenius?

In summary, The conversation revolves around solving a second order linear homogeneous differential equation using the Frobenius method when there is no shift. The participants discuss the lack of information in books and ask for help, eventually figuring out that in this case, the indicial relations act as the two values of the series.
  • #1
fabsuk
51
0
A simple question i think although i can't find in any books

What do u when u are solving a second order linear hoemoeneous differential equation with frobenius and there is no shift.

[tex](X^2)(y^{''}) (-6y)=0[/tex] it should be normal minus -6y

I only know what to do if there is a shift.Help someone?

This is otherwise known as series solutions to differential equations
 
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  • #2
Does anybody know?

I keep trying but get nowhere i get an answer of 0 which i wrong.Help
 
  • #3
Well i think i figured it out.
There is no recurrence relation and so your indical relations become your 2 values of the series.Thanks for the help.:rolleyes:
 

1. How do I determine the characteristic equation for a second order linear homogeneous ODE with Frobenius?

The characteristic equation for a second order linear homogeneous ODE with Frobenius can be determined by substituting the Frobenius series solution into the ODE and equating the coefficients of each power of x to 0.

2. What is the general form of the Frobenius series solution for a second order linear homogeneous ODE?

The general form of the Frobenius series solution for a second order linear homogeneous ODE is given by y(x) = Σn=0∞ an(x-x0)n+r, where an are the coefficients of the series and r is the root of the characteristic equation with multiplicity 2.

3. How do I determine the radius of convergence for the Frobenius series solution?

The radius of convergence for the Frobenius series solution can be determined by using the ratio test. The series will converge when limn→∞ |an+1|/|an| < 1. The radius of convergence is then given by R = 1/limn→∞ |an+1|/|an|.

4. What is the difference between a regular singular point and an irregular singular point in a second order linear homogeneous ODE with Frobenius?

A regular singular point is a point where the Frobenius solution has a finite radius of convergence, while an irregular singular point is a point where the Frobenius solution has a radius of convergence of 0. Regular singular points can be solved using Frobenius method, while irregular singular points require other methods, such as the method of dominant balance.

5. How can I check the accuracy of my Frobenius series solution for a second order linear homogeneous ODE?

You can check the accuracy of your Frobenius series solution by substituting it into the original ODE and comparing the resulting series with the original ODE. If the series are equal, then the solution is accurate. Additionally, you can check the accuracy by using other methods, such as variation of parameters or the Wronskian test.

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