The solution of the differential equation `y^(2)dx+(x^(2)-xy + y^(2))dy = 0` is

Options

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

Answer

A

Explanation

`(dx)/(dy)+(x^(2))/(y^(2))-(x)/(y)+1=0`
Put x = vy
`rArr" "(dx)/(dy)=v+y(dv)/(dy)`
`therefore" "v+y(dv)/(dy)+v^(2)-v+1=0`
`rArr" "-y(dv)/(dy)=(1+v^(2))`
`rArr" "(dv)/(v^(2)+1)=-(dy)/(y)`
Integrating
`tan^(-1) v + C = - "In y"`
`rArr" "tan^(-1)((x)/(y))+ "In y" + C = 0`
where C is arbitrary constant