Classifying Singular Points: Regular or Irregular?

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Homework Statement


Find all singular points of xy"+(1-x)y'+xy=0 and determine whether each one is regular or irregular.


Homework Equations


The answer is x=0, regular.


The Attempt at a Solution


I know that x=0 since you set whatever is in front of y" to 0 and you solve for x, right?
And I think you supposed to take the limit as x approaches to 0 but I don't know which function to take the limit of.
 
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Have you looked in your text to see the definition of a regular singular point? What is it?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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