The solution of the differential equation `(d^2y)/(dx^2)=sin3x+e^x+x^2` when `y_1(0)=1` and `y(0)` is (a) `( b ) (c) (d)(( e )-sin3x)/( f )9( g ) (h)+( i ) e^(( j ) x (k))( l )+( m )(( n ) (o) x^(( p )4( q ))( r ))/( s )(( t ) 12)( u ) (v)+( w )1/( x )3( y ) (z) x-1( a a )` (bb) (cc) `( d d ) (ee) (ff)(( g g )-sin3x)/( h h )9( i i ) (jj)+( k k ) e^(( l l ) x (mm))( n n )+( o o )(( p p ) (qq) x^(( r r )4( s s ))( t t ))/( u u )(( v v ) 12)( w w ) (xx)+( y y )1/( z z )3( a a a ) (bbb) x (ccc)` (ddd) (eee) `( f f f ) (ggg) (hhh)(( i i i )-cos3x)/( j j j )3( k k k ) (lll)+( m m m ) e^(( n n n ) x (ooo))( p p p )+( q q q )(( r r r ) (sss) x^(( t t t )4( u u u ))( v v v ))/( w w w )(( x x x ) 12)( y y y ) (zzz)+( a a a a )1/( b b b b )3( c c c c ) (dddd) x+1( e e e e )` (ffff) (d) None of these

Options

A. `y=(-sin3x)/9 + e^(x)+x^(4)/12+1/3x-1`
B. `y=(-sin3x)/9 + e^(x)+x^(4)/12+1/3x`
C. `y=(-cos3x)/3 + e^(x) + x^(4)/12+1/3x +1`
D. None of these

Answer

A

Explanation

Integrating the given differential equation, we have
`(dy)/(dx) = (-cos3x)/3+e^(x)+x^(3)/3+C_(1)`
But `y_(1)(0)=1`
So, `1=(-1/3)+C_(1)` or `C(1)=(1//3)`
Again integrating, we get
`y=(-sin3x)/9 + e^(x)+x^(4)/12+1/3x+C_(2)`
But y(0)=0, So, 0=0+1 `+C_(2)` or `C_(2)=-1`.
Thus, `y=(-sin3x)/9 + e^(x)+x^(4)/12+1/3x-1`