The solution of the differential equation `(dy)/(dx)=(3x^2y^4+2x y)/(x^2-2x^3y^3)` is (a) `( b ) (c) (d)(( e ) (f) y^(( g )2( h ))( i ))/( j ) x (k) (l)-( m ) x^(( n )3( o ))( p ) (q) y^(( r )2( s ))( t )=c (u)` (v) (w) `( x ) (y) (z)(( a a ) (bb) x^(( c c )2( d d ))( e e ))/( f f )(( g g ) (hh) y^(( i i )2( j j ))( k k ))( l l ) (mm)+( n n ) x^(( o o )3( p p ))( q q ) (rr) y^(( s s )3( t t ))( u u )=c (vv)` (ww) (xx) `( y y ) (zz) (aaa)(( b b b ) (ccc) x^(( d d d )2( e e e ))( f f f ))/( g g g ) y (hhh) (iii)+( j j j ) x^(( k k k )3( l l l ))( m m m ) (nnn) y^(( o o o )2( p p p ))( q q q )=c (rrr)` (sss) (d) `( t t t ) (uuu) (vvv)(( w w w ) (xxx) x^(( y y y )2( z z z ))( a a a a ))/( b b b b )(( c c c c )3y)( d d d d ) (eeee)-2( f f f f ) x^(( g g g g )3( h h h h ))( i i i i ) (jjjj) y^(( k k k k )2( l l l l ))( m m m m )=c (nnnn)` (oooo)

Options

A. `y^(2)/x-x^(3)y^(2)=c`
B. `x^(2)/y^(2)+x^(3)y^(3)=c`
C. `x^(2)/y+x^(3)y^(2)=c`
D. `x^(2)/(3y)-2x^(3)y^(2)=c`

Explanation

To solve the differential equation given by

\[ \frac{dy}{dx} = \frac{3x^2y^4 + 2xy}{x^2 - 2x^3y^3}, \]

we will follow a systematic approach.

### Step 1: Rewrite the Differential Equation

We start by rewriting the equation in a more manageable form. We can rearrange the terms:

\[ 2xy \, dy - x^2 \, dy + (3x^2y^4 + 2xy) \, dx = 0. \]

### Step 2: Divide by \(y^2\)

Next, we divide the entire equation by \(y^2\) to simplify it:

\[ \frac{2x}{y} \, dy - \frac{x^2}{y^2} \, dy + 3x^2 \, dx + 2x^3 \, dy = 0. \]

### Step 3: Identify Integrating Factors

We can observe that the terms can be grouped in a way that suggests a potential integrating factor. We notice that:

\[ \frac{d}{dx}(x^2) = 2x, \]
and
\[ \frac{d}{dx}(x^3y^3) = 3x^2y^3 + x^3 \cdot 3y^2 \cdot \frac{dy}{dx}. \]

### Step 4: Integrate the Equation

Now we can integrate both sides. The left side integrates to:

\[ \int \left( \frac{2x}{y} - \frac{x^2}{y^2} \right) dy = \int 3x^2 \, dx + \int 2x^3 \, dy. \]

This gives us:

\[ x^2 + y + x^3y^3 = C, \]

where \(C\) is the constant of integration.

### Step 5: Final Form of the Solution

Thus, the solution to the differential equation can be expressed as:

\[ x^2 + y + x^3y^3 = C. \]

### Step 6: Check the Options

Now, we need to compare this solution with the provided options to find the correct one.