Power series solutions to differential equations

In summary, the individual is revising and is stuck on question 4 of a specific assignment. They have correctly identified the regular singular points for the first part of the question but are unsure how to approach the next part. They have attempted to use the method of Frobenius and have looked for examples online, but are still struggling. They are seeking guidance to properly solve the question.
  • #1
Kate2010
146
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Homework Statement



I'm revising at the moment and a bit stumped on question 4 http://www.maths.ox.ac.uk/system/files/attachments/AC104.pdf

Homework Equations





The Attempt at a Solution



I think for the first part of the question, the regular singular points are 0 and -2.

However, I am unsure as to how to tackle the next part. I have assumed it is ok to differentiate the power series term by term and have done so and subbed it back into the original equation, now I feel like I need to equate coefficients, but I feel like I have no idea what I'm doing. If you could point me in the right direction I'd be really grateful :).
 
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  • #2
This is the method of Frobenius, there are examples everywhere on the net.
 
  • #3
Thanks :) I get it now.
 

FAQ: Power series solutions to differential equations

What is a power series solution to a differential equation?

A power series solution to a differential equation is a method of solving a differential equation by expressing the solution as a series of terms, each containing a variable raised to a different power. This allows for an infinite number of terms to be used, providing a more accurate solution than traditional methods.

When is a power series solution useful?

A power series solution is particularly useful when the differential equation cannot be solved using other methods, or when an exact solution is not necessary. It is also useful in situations where a numerical solution is required, as it can provide a more accurate result than using a finite number of terms.

How do you find the coefficients of a power series solution?

The coefficients of a power series solution can be found by substituting the series into the differential equation and equating coefficients of like powers. This results in a system of equations that can be solved to determine the values of the coefficients.

What is the radius of convergence for a power series solution?

The radius of convergence is the maximum distance from the center of the series at which the series will converge. For a power series solution, the radius of convergence depends on the coefficients of the series and the differential equation being solved. It is important to check the radius of convergence to ensure that the series provides an accurate solution.

Can a power series solution be used for all types of differential equations?

No, a power series solution is not applicable for all types of differential equations. It is typically used for linear differential equations with variable coefficients. It may also be used for some nonlinear equations, but the process becomes more complicated and the accuracy of the solution may be reduced.

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