The differential equation of all parabolas whose axis are parallel to the y-axis is (a) `( b ) (c) (d)(( e ) (f) d^(( g )3( h ))( i ) y)/( j )(( k ) d (l) x^(( m )3( n ))( o ))( p ) (q)=0( r )` (s) (b) `( t ) (u) (v)(( w ) (x) d^(( y )2( z ))( a a ) x)/( b b )(( c c ) d (dd) y^(( e e )2( f f ))( g g ))( h h ) (ii)=C (jj)` (kk) (c) `( d ) (e) (f)(( g ) (h) d^(( i )3( j ))( k ) y)/( l )(( m ) d (n) x^(( o )3( p ))( q ))( r ) (s)+( t )(( u ) (v) d^(( w )2( x ))( y ) x)/( z )(( a a ) d (bb) y^(( c c )2( d d ))( e e ))( f f ) (gg)=0( h h )` (ii) (d) `( j j ) (kk) (ll)(( m m ) (nn) d^(( o o )2( p p ))( q q ) y)/( r r )(( s s ) d (tt) x^(( u u )2( v v ))( w w ))( x x ) (yy)+2( z z )(( a a a ) dy)/( b b b )(( c c c ) dx)( d d d ) (eee)=C (fff)` (ggg)

Options

A. `(d^(3)y)/(dx^(3)) =0`
B. `(d^(2)x)/(dy^(2)) =C`
C. ` (d^(3) x)/(dy^(3)) =1`
D. `(d^(3)y)/(dx^(3)) =C`

Answer

A

Explanation

The general equation of all parabolas where axes are parallel to Y -axis is
` y= Ax^(2) +Bx +C`
Where A,B and C are arbitaray constants on differentiating eq. (i) we w.r.t.s we get
`(dy)/(dx) =2Ax +B`
On differentiating eq (ii) w.r.t.x we get
`(d^(2)y)/(dx^(2)) = 2A`
On differentiating eq. (iii) w.r.t.x, we get
` (d^(3)y)/(dx^(3)) =0`