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Math Help - Regular Singular Points

  1. #1
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    Regular Singular Points

    For those that are interested, the problem has been taken from the Elementary Differential Equations and Boundary Value Problems, 8th Edition, by Boyce. It is problem 1 from Section 5.4.

    The problem states to find all singular points of the given equation and determine whether each one is regular or irregular.

     <br />
x y'' + (1 - x) y' + x y = 0 <br />

     <br />
P(x) = x = 0 <br />

    The point x = 0 is a singular point.

    So all other points are ordinary points.

    <br />
\lim_{x \to 0} (x - 0) \frac {(1 - x)} {(x)} = 1<br />

    <br />
\lim_{x \to 0} (x - 0)^2 \frac {(x)} {(x)} = 0<br />

    So as my conclusion the two limits are finite numbers and we can say that the point x = 0 is a regular singular point
    Last edited by fudawala; July 8th 2007 at 11:51 AM. Reason: Made some errors on the Latex Program
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by fudawala View Post
    ...
     <br />
P(x) = x = 0 <br />
    ...
    yes, correct

    the above statement is awkward though

    you should have said something like:

    xy'' + (1 - x)y' + xy = 0

    \Rightarrow y'' + \frac {1 - x}{x} y' + y = 0

    we have that the function \frac {1 - x}{x} is discontinuous at the point x = 0

    thus x = 0 is a singular point

    ...and continue from there
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