The solution of `(dy)/(dx) = (x^(2)+y^(2)+1)/(2xy)` satisfying y(1) =1 is given by

Options

A. a hyperbola
B. a circle
C. `y^(2)=x(1+x)-10`
D. `(x-2)^(2)+(y-3)^(2)=5`

Answer

A

Explanation

We have,
`(dy)/(dx)=(x^(2)+y^(2)+1)/(2xy)`
`rArr" "2xy dy=(x^(2)+y^(2)+1)dx`
`rArr" "2xydy-y^(2)dx=(x^(2)+1)dx`
`rArr" "xd(y^(2))-y^(2)dx=(x^(2)+1)dx`
`rArr" "(xd(y^(2))-y^(2)dx)/(x^(2))=(1+(1)/(x^(2)))dx`
`rArr" "d((y^(2))/(x))=d(x-(1)/(x))`
On integrating, we get
`(y^(2))/(x)=x-(1)/(x)+C`
`rArr" "y^(2)=x^(2)-1+Cx rArr y^(2)=(x+(C)/(2))^(2)-1-(C^(2))/(4)`
Clearly, it represents a hyperbola.