Solve Differential Equation with Notation: Reduction of Order Help

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I need to solve the following differential equation, And am pretty sure it will require the use of Reduction of Order but have NO clue how do deal with the notation on the RH side, any help Would Be Greatly appreciated.
 

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What's the question about the notation on the RH side? It looks clear enough to me. It's the derivative of y squared over x times (-1). And it does look like substituting y'=u is a good reduction of order first step. So y''=u'. Please continue.
 
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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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