The solution of the differential equation `{1+xsqrt((x^2+y^2))}dx+{sqrt((x^2+y^2))-1}ydy=0` is equal to (a) `( b ) (c) (d) x^(( e )2( f ))( g )+( h )(( i ) (j) y^(( k )2( l ))( m ))/( n )2( o ) (p)+( q )1/( r )3( s ) (t) (u) (v)(( w ) (x) (y) x^(( z )2( a a ))( b b )+( c c ) y^(( d d )2( e e ))( f f ) (gg))^(( h h ) (ii) (jj)3/( k k )2( l l ) (mm) (nn))( o o )=c (pp)` (qq) (rr) `( s s ) (tt) x+( u u )(( v v ) (ww) y^(( x x )3( y y ))( z z ))/( a a a )3( b b b ) (ccc)+( d d d )1/( e e e )2( f f f ) (ggg) (hhh) (iii)(( j j j ) (kkk) (lll) x^(( m m m )2( n n n ))( o o o )+( p p p ) y^(( q q q )2( r r r ))( s s s ) (ttt))^(( u u u ) (vvv) (www)1/( x x x )2( y y y ) (zzz) (aaaa))( b b b b )=c (cccc)` (dddd) (eeee) `( f f f f ) (gggg) x-( h h h h )(( i i i i ) (jjjj) y^(( k k k k )2( l l l l ))( m m m m ))/( n n n n )2( o o o o ) (pppp)+( q q q q )1/( r r r r )3( s s s s ) (tttt) (uuuu) (vvvv)(( w w w w ) (xxxx) (yyyy) x^(( z z z z )2( a a a a a ))( b b b b b )+( c c c c c ) y^(( d d d d d )2( e e e e e ))( f f f f f ) (ggggg))^(( h h h h h ) (iiiii) (jjjjj)3/( k k k k k )2( l l l l l ) (mmmmm) (nnnnn))( o o o o o )=c (ppppp)` (qqqqq)

Options

A. `x^(2)+y^(2)/2+1/3(x^(2)+y^(2))^(3//2)=c`
B. `x-y^(3)/3+1/2(x^(2)+y^(2))^(1//2)=c`
C. `x-y^(2)/2+1/3(x^(2)+y^(2))^(3//2)=c`
D. None of these

Answer

C

Explanation

`{1+xsqrt((x^(2)+y^(2)))}dx+{sqrt((x^(2)+y^(2))-1)}ydy=0`
or `dx-ydy+sqrt((x^(2)+y^(2))(xdx+ydy))=0`
or `dx-ydy 1/2sqrt((x^(2)+y^(2))d(x^(2)+y^(2)))=0`
Integrating, we have
`x-y^(2)/2+1/2int(sqrt(t)dt) = c, {t=sqrt((x^(2)+y^(2)))}`
Or `x-y^(2)/2+1/3(x^(2)+y^(2))^(3//2)=c`