Series Solution around singular point

In summary, the conversation discusses finding the general solution of the given differential equation using a polynomial and a series in powers of x - 1. The suggested approach is to change variables to simplify the algebra and it is questioned whether a polynomial solution is possible.
  • #1
hadroneater
59
0

Homework Statement



x(2 - x)y'' - (x - 1)y' + 2y = 0
Find the general solution in terms of a polynomial and a series in powers of x - 1.

Homework Equations


The Attempt at a Solution


Does the question basically ask for a series solution of the ODE at the regular point x = 1?
Then [itex]y(x) = \sum^{∞}_{n = 0}c_{n}(x - 1)^{n}[/itex]
If I sub that into the ODE then I get a rather complicated algebraic mess for x(2 - x)y''. Is this the right way to solve this?
 
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  • #2
Try changing variables from x to u=x-1. That'll simplify the algebra a bit.

I don't see how you're going to get a polynomial solution. Did you type the DE as given?
 
Last edited:

What is a singular point in a series solution?

A singular point in a series solution is a point in the function where the function is not defined or becomes infinite.

How do you find the series solution around a singular point?

To find the series solution around a singular point, you first need to identify the singular point in the function. Then, you can use techniques such as power series or Frobenius method to expand the function into an infinite series.

What is the significance of finding a series solution around a singular point?

Finding a series solution around a singular point allows us to approximate the behavior of a function near the singular point. This can be useful in various applications, such as in engineering or physics, where understanding the behavior of a function near a singularity is important.

Can series solutions around singular points be used to find exact solutions?

No, series solutions around singular points are typically used to approximate the behavior of a function near a singularity. They cannot be used to find exact solutions as they involve an infinite number of terms and may only provide an approximation of the function.

Are there any limitations to using series solutions around singular points?

Yes, there are limitations to using series solutions around singular points. These solutions may only be valid in a certain region around the singular point and may not accurately represent the behavior of the function outside of that region. Additionally, the accuracy of the approximation may decrease with increasing distance from the singular point.

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