Series solution to 2nd order differential equation

In summary, it is not always necessary to shift the index to 2 and 1 when using the power series to solve an ODE. It can still work out fine if the index is left at n=0. The frobenius method can be used for a differential equation about a regular singular point, which falls under the category of using series solutions to solve 2nd order ODEs.
  • #1
drsmoothe2004
7
0
when using the power series to solve an ODE, is it always necessary to shift the index to 2 and 1 when taking the second and first derivatives of the power series respectively?

eq0035MP.gif


i noticed that if i don't shift the index at all and leave them at n=0, it still works out fine?

also, how will i know when to use the frobenius method vs the power series solution?

thirdly, frobenius and power series fall under the category of how to use the series solution to solve 2nd order ODEs right?
 

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  • #2
Hello,

I can answer the third question.

I my book "A First Course In Differential Equations" 8th Edition by Zill it is stated

The Method of Frobenius can be used to a differential equation about a regular singular point. The differential equation is to be of this form

a2(x)y''+a1(x)y'+a0(x)y=0

So yes, you are correct in stating that Frobenius can be used to find a series solution of a 2nd order ODE.

Thanks
Matt
 

Related to Series solution to 2nd order differential equation

1. What is a series solution to a 2nd order differential equation?

A series solution to a 2nd order differential equation is a method of solving a differential equation by expanding the solution in a series of terms. This allows for an approximate solution to be found, as the series is truncated at a certain point.

2. How does a series solution differ from other methods of solving differential equations?

A series solution differs from other methods, such as separation of variables or integrating factors, in that it uses a series expansion to approximate the solution instead of finding an exact analytical solution. This is useful for more complex equations that cannot be solved using traditional methods.

3. What is the process for finding a series solution to a 2nd order differential equation?

The process for finding a series solution involves rewriting the differential equation in terms of the independent variable and its derivatives, then substituting this into a power series expression for the solution. This will result in a recurrence relation that can be used to find the coefficients of the series.

4. When is a series solution the most appropriate method for solving a 2nd order differential equation?

A series solution is most appropriate when the equation cannot be solved using traditional methods or when an approximate solution is sufficient. It is also useful when the equation has variable coefficients or when the boundary conditions are difficult to work with.

5. Are there any limitations to using a series solution for 2nd order differential equations?

Yes, there are limitations to using a series solution. In some cases, the series may not converge or may converge too slowly to be useful. This can occur when the solution has a singularity or when the coefficients in the equation grow too quickly. In these cases, other methods may be more appropriate.

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