Homogenous differential equation

rbailey5
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1. Homework Statement [/b

so I am given a known homogeneous differential equation dy/dx=(y^2+x*sqrt(x^2+y^2))/xy


Homework Equations


now I know that you have to separate into some form of y/x which then you can change into v and solve the differential equation but I am having trouble


The Attempt at a Solution


I just can't figure out how to get it in the appropriate form in order to simplify
 
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Let y = ux, y' = u'x + u. Work it out and show us what happens.
 

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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