Find the general solution of `y^(2)dx+(x^(2)-xy+y^(2))dy=0`

Options

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

Answer

D

Explanation

We have,
`y^(2)dx+(x^(2)-xy+y^(2))dy=0rArr (dy)/(dx)=-(y^(2))/(x^(2)-xy+y^(2))`
Putting `y=vx and (dy)/(dx)=v+x(dv)/(dx)`, we get
`v+x(dv)/(dx)=-(v^(2))/(1-v+v^(2))`
`rArr" "x(dv)/(dx)=(-v-v^(3))/(v^(2)-v+1)`
`rArr" "(v^(2)-v+1)/(v(v^(2)+1))dv=-(dx)/(x)`
`rArr" "((1)/(x)-(1)/(v^(2)+1))dv=-(dx)/(x)`
On integrating, we get
`logv-tan^(-1)v=-logx+C`
`rArr" "log((y)/(x))-tan^(-1).(y)/(x)=-logx+CrArr log y = tan^(-1).(y)/(x)+C`