Solving the Homogeneous Equation y2dx -x(2x+3y)dy =0

mamma_mia66
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Homework Statement


y2dx -x(2x+3y)dy =0 I have to recognize the equation and solve it


Homework Equations





The Attempt at a Solution



I did y2dx - (2x2+ 3yx) dy=0
which is a homogeneous now
after I substitude x=uy
dx=udy + ydu

I stuck here after the substitution

y2udy+y3du - 2x2dy + 3yxdy=0

please someone help if I am not in right direction from the begining.
 
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mamma_mia66 said:

Homework Statement


y2dx -x(2x+3y)dy =0 I have to recognize the equation and solve it


Homework Equations





The Attempt at a Solution



I did y2dx - (2x2+ 3yx) dy=0
which is a homogeneous now
after I substitude x=uy
dx=udy + ydu

I stuck here after the substitution

y2udy+y3du - 2x2dy + 3yxdy=0

please someone help if I am not in right direction from the begining.
I am more accustomed to write y= ux but your method is equivalent.
Saying x= uy is the same as saying u= y/x. Divide your last equation by x^2 and see what you get!
 
ok, thanks.
 

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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