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

Explanation

To solve the differential equation

\[ \frac{d^2y}{dx^2} = \sin(3x) + e^x + x^2 \]

with the initial conditions \( y'(0) = 1 \) and \( y(0) = 0 \), we will follow these steps:

### Step 1: Integrate the differential equation

Integrate both sides with respect to \( x \):

\[ \frac{dy}{dx} = \int \left( \sin(3x) + e^x + x^2 \right) dx \]

### Step 2: Calculate the integrals

The integrals can be calculated as follows:

- The integral of \( \sin(3x) \) is \( -\frac{1}{3} \cos(3x) \).
- The integral of \( e^x \) is \( e^x \).
- The integral of \( x^2 \) is \( \frac{x^3}{3} \).

Thus, we have:

\[ \frac{dy}{dx} = -\frac{1}{3} \cos(3x) + e^x + \frac{x^3}{3} + C_1 \]

where \( C_1 \) is a constant of integration.

### Step 3: Integrate again to find \( y \)

Now, integrate \( \frac{dy}{dx} \):

\[ y = \int \left( -\frac{1}{3} \cos(3x) + e^x + \frac{x^3}{3} + C_1 \right) dx \]

Calculating these integrals:

- The integral of \( -\frac{1}{3} \cos(3x) \) is \( -\frac{1}{9} \sin(3x) \).
- The integral of \( e^x \) is \( e^x \).
- The integral of \( \frac{x^3}{3} \) is \( \frac{x^4}{12} \).

Thus, we have:

\[ y = -\frac{1}{9} \sin(3x) + e^x + \frac{x^4}{12} + C_1 x + C_2 \]

where \( C_2 \) is another constant of integration.

### Step 4: Apply initial conditions

Now, we apply the first initial condition \( y'(0) = 1 \):

Substituting \( x = 0 \):

\[ 1 = -\frac{1}{3} \cos(0) + e^0 + 0 + C_1 \]

This simplifies to:

\[ 1 = -\frac{1}{3} + 1 + C_1 \]

Thus:

\[ C_1 = \frac{1}{3} \]

Now, apply the second initial condition \( y(0) = 0 \):

Substituting \( x = 0 \):

\[ 0 = -\frac{1}{9} \sin(0) + e^0 + 0 + 0 + C_2 \]

This simplifies to:

\[ 0 = 1 + C_2 \]

Thus:

\[ C_2 = -1 \]

### Step 5: Write the final solution

Substituting \( C_1 \) and \( C_2 \) back into the equation for \( y \):

\[ y = -\frac{1}{9} \sin(3x) + e^x + \frac{x^4}{12} + \frac{1}{3} x - 1 \]

### Final Answer

Thus, the solution to the differential equation is:

\[ y = -\frac{1}{9} \sin(3x) + e^x + \frac{x^4}{12} + \frac{1}{3} x - 1 \]