The general solution of
`y^(2)dx+(x^(2)-xy+y^(2))dy=0`, is

Options

A. `tan^(-1) (y/x) = log xy + c`
B. ` 2 tan^(-1) (x/y) + log x + c=0`
C. ` log y + sqrt(x^(2)+y^(2)) + log y +c =0`
D. `sinh^(-1) (x/y) + log y +c =0`

Answer

A

Explanation

Given ` (dy)/(dx) = (-y^(2))/(x^(2)-xy+y^(2))`
Put y =ux
` Rightarrow (dy)/(dx) = v + x(dv)/(dx) `
` v +x(dv)/(dx) = (-v^(2))/(v^(2)-v+1)`
` x (dv)/(dx) = (-v^(3)-v)/(v^(2) -v+1)`
` Rightarrow ((v^(2) -v+1))/(-v^(3)-v) dv = 1/x dx `
`Rightarrow (-(v^(2)+1)+v)/(v(v^(2)+1)) dv = 1/x dx`
` Rightarrow (-v^(2)+1)+v)/(v(v^(2)+1)) dv = 1/x dx`
` Rightarrow int - 1/v dv + int 1/(v^(2)+1) dv = int 1/x dx`
` Rightarrow -log v + tan^(-1) v = log x +c `
` Rightarrow tan^(-1) v = log xv +c `
` Rightarrow tan^(-1) (y/x) = log y +c `