Question #179783

solve dx/(x-y)y^2 = dy/(y-x)x^2 = dz/x^2+y^2


1
Expert's answer
2021-04-29T16:47:27-0400




The question is

\bigstar

dx((xy)y2)dx\over { ((x-y)y^2) } =dy((yx)x2)dy\over((y-x)x^2) = dz(x2+y2)dz\over(x^2+y^2)



Consider these



dx((xy)y2)dx\over { ((x-y)y^2) } =dy((yx)x2)dy\over((y-x)x^2)


We get

Integration of

\bull

y2dy=x2dx\boxed{\int y^2dy = \int -x^2dx}



\bull

y3=x3+c1\boxed{y^3 = -x^3+c1}




\bigstar

dx((xy)y2)=dz(x2+y2){dx\over ((x-y)y^2)}={dz\over(x^2+y^2)}



     \implies


y2dz=(x2+y2)(xy)dx\boxed{ \int {y^2} dz= \int {(x^2+y^2)\over (x-y) }dx }


    \implies


(x2+y2)(xy)dx\boxed{∫ {(x^2+y^2)\over (x-y) }dx} =2y2(xy)+x+y)dx\boxed{∫ {2y^2\over(x-y)+x+y)}dx} =




2y21(xy)dx+xdx+y1dx=\boxed{2y^2\intop{1\over(x-y)}dx +\intop x dx + y \intop 1 dx}=


= 2y2ln(xy)+x22+xy+c2\boxed{2 y^2 ln(x-y) + {x^2\over{2 }}+ xy + c_2}



\bull

\bigstar

y2ln(z)=2y2ln(xy)+x22+xy+c2\boxed{y^2ln(z) = 2 y^2 ln(x-y) + {x^2\over{2 }}+ xy + c_2 }


integral surface:

\bull

y3=x3+c1\boxed{ y^3 = -x^3+c_1 }



\bull

y2ln(z)=2y2ln(xy)+x22+xy+c2\boxed{y^2ln(z) = 2 y^2 ln(x-y) + {x^2\over{2 }}+ xy + c_2 }



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS