The solution of differential equation `x y^(prime)=x((y^2)/(x^2)+(f((y^2)/(x^2)))/(f^(prime)((y^2)/(x^2))))` is (a) `( b ) (c)f(( d ) (e) (f)(( g ) (h) y^(( i )2( j ))( k ))/( l )(( m ) (n) x^(( o )2( p ))( q ))( r ) (s) (t))=c (u) x^(( v )2( w ))( x ) (y)` (z) (b) `( a a ) (bb) (cc) x^(( d d )2( e e ))( f f )f(( g g ) (hh) (ii)(( j j ) (kk) y^(( l l )2( m m ))( n n ))/( o o )(( p p ) (qq) x^(( r r )2( s s ))( t t ))( u u ) (vv) (ww))=( x x ) c^(( y y )2( z z ))( a a a ) (bbb) y^(( c c c )2( d d d ))( e e e ) (fff)` (ggg) (c) `( d ) (e) (f) x^(( g )2( h ))( i )f(( j ) (k) (l)(( m ) (n) y^(( o )2( p ))( q ))/( r )(( s ) (t) x^(( u )2( v ))( w ))( x ) (y) (z))=c (aa)` (bb) (d) `( c c ) (dd)f(( e e ) (ff) (gg)(( h h ) (ii) y^(( j j )2( k k ))( l l ))/( m m )(( n n ) (oo) x^(( p p )2( q q ))( r r ))( s s ) (tt) (uu))=( v v )(( w w ) c y)/( x x ) x (yy) (zz) (aaa)` (bbb)

Options

A. `f(y^(2)//x^(2))=cx^(2)`
B. `x^(2)f(y^(2)//x^(2))=c^(2)y^(2)`
C. `x^(2)f(y^(2)//x^(2))=c`
D. `f(y^(2)//x^(2))=cy//x`

Answer

A

Explanation

The given equation can be written as
`y/x(dy)/(dx) ={y^(2)/x^(2)+(f(y^(2)//x^(2))/f^(')(y^(2)//x^(2)))}`
The above equation is a homogenous equation.
Putting `y=vx`, we get
`v[v+x(dv)/(dx)]=v^(2)+f(v^(2))/f^(')(v^(2))` (variable separable)
or `(2vf^(')(v^(2)))/(f(v^(2)))dv=2(dx)/x`
Now, integrating both sides, we get
`logf(v^(2))=logx^(2)+logc` [logc= constant]
or `logf(v^(2))=logcx^(2)`
or `f(v^(2))=cx^(2)`
or `f(y^(2)//x^(2))=cx^(2)`