The solution of `(dy)/(dx)=(x^2+y^2+1)/(2x y)` satisfying `y(1)=1` is given by (a) a system of parabolas (b) a system of circles (c) `( d ) (e) (f) y^(( g )2( h ))( i )=x(( j ) (k)1+x (l))-1( m )` (n) (d) `( o ) (p) (q) (r)(( s ) (t) x-2( u ))^(( v )2( w ))( x )+( y ) (z)(( a a ) (bb) y-3( c c ))^(( d d )2( e e ))( f f )=5( g g )` (hh)

Options

A. a system of parabolas
B. a system of circles
C. `y^(2)=x(1+x)-1`
D. `(x-2)^(2)+(y-3)^(2)=5`

Answer

C

Explanation

Rewritting the given equation is
`2xy(dy)/(dx) -y^(2)=1+x^(2)`
or `2y(dy)/(dx)-1/xy^(2)=1/x+x`
Putting `y^(2)=u`, we have
`(du)/(dx) -1/xu=1/x+x`
I.F. `=e^(-int1/xdx)=1/x`
Thus, solution is `u1/x=int(1/x^(2)+1)dx=-1/x+x+C`
or `y^(2)=(x^(2)-1)+Cx`
Since `y(1)=1`, we get C=1.
Hence, `y^(2)=x(1+x)-1` which represents a system of hyperbola.