The differential equation whose general solution is given by `y=(c_1cos(x+c_2)-(c_3e^((-x+c4))+(c_5sinx),` where `c_1,c_2,c_3,c_4,c_5` are arbitrary constants, is (a) `( b ) (c) (d)(( e ) (f) d^(( g )4( h ))( i ) y)/( j )(( k ) d (l) x^(( m )4( n ))( o ))( p ) (q)-( r )(( s ) (t) d^(( u )2( v ))( w ) y)/( x )(( y ) d (z) x^(( a a )2( b b ))( c c ))( d d ) (ee)+y=0( f f )` (gg) (hh) `( i i ) (jj) (kk)(( l l ) (mm) d^(( n n )3( o o ))( p p ) y)/( q q )(( r r ) d (ss) x^(( t t )3( u u ))( v v ))( w w ) (xx)+( y y )(( z z ) (aaa) d^(( b b b )2( c c c ))( d d d ) y)/( e e e )(( f f f ) d (ggg) x^(( h h h )2( i i i ))( j j j ))( k k k ) (lll)+( m m m )(( n n n ) dy)/( o o o )(( p p p ) dx)( q q q ) (rrr)+y=0( s s s )` (ttt) (uuu) `( v v v ) (www) (xxx)(( y y y ) (zzz) d^(( a a a a )5( b b b b ))( c c c c ))/( d d d d )(( e e e e ) d (ffff) x^(( g g g g )5( h h h h ))( i i i i ))( j j j j ) (kkkk)+y=0( l l l l )` (mmmm) (nnnn) `( o o o o ) (pppp) (qqqq)(( r r r r ) (ssss) d^(( t t t t )3( u u u u ))( v v v v ) y)/( w w w w )(( x x x x ) d (yyyy) x^(( z z z z )3( a a a a a ))( b b b b b ))( c c c c c ) (ddddd)-( e e e e e )(( f f f f f ) (ggggg) d^(( h h h h h )2( i i i i i ))( j j j j j ) y)/( k k k k k )(( l l l l l ) d (mmmmm) x^(( n n n n n )2( o o o o o ))( p p p p p ))( q q q q q ) (rrrrr)+( s s s s s )(( t t t t t ) dy)/( u u u u u )(( v v v v v ) dx)( w w w w w ) (xxxxx)-y=0( y y y y y )` (zzzzz)

Options

A. `(d^(4)y)/(dx^(4)) - (d^(2)y)/(dx^(2))+y=0`
B. `(d^(3)y)/(dx^(3))+(d^(2)y)/(dx^(2))+(dy)/(dx)+y=0`
C. `(d^(5)y)/(dx^(5))+y=0`
D. `(d^(3)y)/(dx^(3))-(d^(2)y)/(dx^(2))+(dy)/(dx)-y=0`

Answer

B

Explanation

`y=c_(1)cos(x+c_(2))-(c_(3)e^(-x+c_(4)))+(c_(5)sinx)`
`=c_(1)(cosxcosc_(2))cosx-(c_(1)sinc_(2)-c_(5))sinx-(c_(3)e^(c^(4))e^(-x))+(c_(5)sinx)`
`=(c_(1)cos c_(2)) cosx-(c_(1)sinc_(2)-c_(5))sinx-(c_(3)e^(c^(4)))e^(-x)`
or `y=lcosx + msin x-ne^(-x)`............(1)
where l,m,n are arbitary constants
`therefore (dy)/(dx)=-lsinx+mcosx+me^(x)`.............(2)
or `(d^(2)y)/(dx^(2))=-lcosx-msinx-ne^(-x)`.............(3)
or `(d^(3)y)(dx^(3))=lsinx-mcosx+ne^(-x)`............(4)
From equations (1) + (3) , `(d^(2)y)/(dx^(2))+y=-2ne^(-x)`............(5)
From equations (2) +(4), `(d^(3)y)/(dx^(3))+(dy)/(dx)=2ne^(-x)`.............(6)
From equations (5) + (6), `(d^(3)y)/(dx^(3))+(dy)/(dx)+y=0`