What's the difference between these two D.E?

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1. x(x-1)y``+(3x-1)y`+y=0
2. (x^2 -x)y``-xy`+y=0
those are 2 examples of the frobenius method, but i don't understand why those 2 use completely different methods to solve...
 
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What methods were used to solve each of them?
 
the first used the method of variation of parameters, and the second one
supposed that y2=cy1 and differentiated that and subsituted y2``, y` and y into the original equation...
 
Firstly, can you stop calling DEs frobenius: they are not. Frobenius refers to the use of a power series solution for DEs, there are different ways to work out what that power series is.
 

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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