The differential equation of the family of curves `y=e^x(Acosx+Bsinx),` where `A` and `B` are arbitrary constants is (a) `( b ) (c) (d)(( e ) (f) d^(( g )2( h ))( i ) y)/( j )(( k ) d (l) x^(( m )2( n ))( o ))( p ) (q)-2( r )(( s ) dy)/( t )(( u ) dx)( v ) (w)+2y=0( x )` (y) (z) `( a a ) (bb) (cc)(( d d ) (ee) d^(( f f )2( g g ))( h h ) y)/( i i )(( j j ) d (kk) x^(( l l )2( m m ))( n n ))( o o ) (pp)+2( q q )(( r r ) dy)/( s s )(( t t ) dx)( u u ) (vv)+2y=0( w w )` (xx) (yy) `( z z ) (aaa) (bbb)(( c c c ) (ddd) d^(( e e e )2( f f f ))( g g g ) y)/( h h h )(( i i i ) d (jjj) x^(( k k k )2( l l l ))( m m m ))( n n n ) (ooo)+( p p p ) (qqq)(( r r r ) (sss) (ttt)(( u u u ) dy)/( v v v )(( w w w ) dx)( x x x ) (yyy) (zzz))^(( a a a a )2( b b b b ))( c c c c )+y=0( d d d d )` (eeee) (ffff) `( g g g g ) (hhhh) (iiii)(( j j j j ) (kkkk) d^(( l l l l )2( m m m m ))( n n n n ) y)/( o o o o )(( p p p p ) d (qqqq) x^(( r r r r )2( s s s s ))( t t t t ))( u u u u ) (vvvv)-7( w w w w )(( x x x x ) dy)/( y y y y )(( z z z z ) dx)( a a a a a ) (bbbbb)+2y=0( c c c c c )` (ddddd)

Options

A. `(d^(2)y)/(dx^(2))-2(dy)/(dx)+2y=0`
B. `(d^(2)y)/(dx^(2))+2(dy)/(dx)-2y=0`
C. `(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)+y=0`
D. `(d^(2)y)/(dx^(2))-7(dy)/(dx)+2y=0`

Answer

A

Explanation

`y=e^(x)(A cosx+Bsinx)`
`(dy)/(dx) = e^(x)[-Asinx+Bcosx]+e^(x)[A cosx+Bsinx]`
`(dy)/(dx) = e^(x)[-A sinx + Bcosx]+y` ……………(1)
Again, differentiating w.r.t x, we ger
`(d^(2)y)/(dx^(2))=(dy)/(dx)-y-y+(dy)/(dx)` [using 1]
`(d^(2)y)/(dx)-2(dy)/(dx)+2y=0`