The solution of the differential equation `(dy)/(dx)=(3x^2y^4+2x y)/(x^2-2x^3y^3)` is (a) `( b ) (c) (d)(( e ) (f) y^(( g )2( h ))( i ))/( j ) x (k) (l)-( m ) x^(( n )3( o ))( p ) (q) y^(( r )2( s ))( t )=c (u)` (v) (w) `( x ) (y) (z)(( a a ) (bb) x^(( c c )2( d d ))( e e ))/( f f )(( g g ) (hh) y^(( i i )2( j j ))( k k ))( l l ) (mm)+( n n ) x^(( o o )3( p p ))( q q ) (rr) y^(( s s )3( t t ))( u u )=c (vv)` (ww) (xx) `( y y ) (zz) (aaa)(( b b b ) (ccc) x^(( d d d )2( e e e ))( f f f ))/( g g g ) y (hhh) (iii)+( j j j ) x^(( k k k )3( l l l ))( m m m ) (nnn) y^(( o o o )2( p p p ))( q q q )=c (rrr)` (sss) (d) `( t t t ) (uuu) (vvv)(( w w w ) (xxx) x^(( y y y )2( z z z ))( a a a a ))/( b b b b )(( c c c c )3y)( d d d d ) (eeee)-2( f f f f ) x^(( g g g g )3( h h h h ))( i i i i ) (jjjj) y^(( k k k k )2( l l l l ))( m m m m )=c (nnnn)` (oooo)

Options

A. `y^(2)/x-x^(3)y^(2)=c`
B. `x^(2)/y^(2)+x^(3)y^(3)=c`
C. `x^(2)/y+x^(3)y^(2)=c`
D. `x^(2)/(3y)-2x^(3)y^(2)=c`

Answer

C

Explanation

Rewrite the differential equation as
`(2xydx-x^(2)dy)+y^(2)(3x^(2)y^(2)dx+2x^(3)ydy)=0`
Dividing by `y^(3)`, we get
`(y^(2)xdx-x^(2)dy)/(y^(2))=y^(2)3x^(2)dx+x^(3)2ydy=0`
or `d(x^(2)/y)+d(x^(3)y^(3))=0`
Integrating, we get the solution
`x^(2)/y+x^(3)y^(2)=c`