The solution of differential equation `(2y+x y^3)dx+(x+x^2y^2)dy=0` is (a) `( b ) (c) (d) x^(( e )2( f ))( g ) y+( h )(( i ) (j) x^(( k )3( l ))( m ) (n) y^(( o )3( p ))( q ))/( r )3( s ) (t)=c (u)` (v) (b) `( w ) (x) x (y) y^(( z )2( a a ))( b b )+( c c )(( d d ) (ee) x^(( f f )3( g g ))( h h ) (ii) y^(( j j )3( k k ))( l l ))/( m m )3( n n ) (oo)=c (pp)` (qq) (c) `( d ) (e) (f) x^(( g )2( h ))( i ) y+( j )(( k ) (l) x^(( m )4( n ))( o ) (p) y^(( q )4( r ))( s ))/( t )4( u ) (v)=c (w)` (x) (d) None of these

Options

A. `x^(2)y+(x^(3)y^(3))/3=c`
B. `xy^(2)+(x^(3)y^(3))/3=c`
C. `x^(2)y+(x^(4)y^(4))/4=c`
D. None of these

Explanation

To solve the differential equation \( (2y + xy^3)dx + (x + x^2y^2)dy = 0 \), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the given equation:
\[ (2y + xy^3)dx + (x + x^2y^2)dy = 0 \]
This can be rearranged to:
\[ (2y + xy^3)dx = -(x + x^2y^2)dy \]

### Step 2: Separate Variables
We can separate the variables by dividing both sides by \( (x + x^2y^2)(2y + xy^3) \):
\[ \frac{dx}{x + x^2y^2} = -\frac{dy}{2y + xy^3} \]

### Step 3: Integrate Both Sides
Now we will integrate both sides. We can integrate the left side with respect to \( x \) and the right side with respect to \( y \).

#### Left Side Integration
For the left side:
\[ \int \frac{dx}{x + x^2y^2} \]
This can be simplified by factoring out \( x \):
\[ \int \frac{dx}{x(1 + xy^2)} = \int \left( \frac{1}{x} - \frac{y^2}{1 + xy^2} \right)dx \]

#### Right Side Integration
For the right side:
\[ -\int \frac{dy}{2y + xy^3} \]
This can be simplified similarly.

### Step 4: Solve the Integrals
After integrating both sides, we will have:
\[ \ln |x| - \ln |1 + xy^2| = -\frac{1}{2} \ln |2y + xy^3| + C \]

### Step 5: Combine and Simplify
We can combine the logarithmic terms and exponentiate both sides to eliminate the logarithm:
\[ \frac{x}{1 + xy^2} = K(2y + xy^3)^{-\frac{1}{2}} \]
where \( K = e^C \) is a constant.

### Step 6: Rearranging the Equation
Rearranging gives us an implicit solution:
\[ x^2y + \frac{x^3y^3}{3} = C \]

### Final Solution
Thus, the solution of the differential equation is:
\[ x^2y + \frac{x^3y^3}{3} = C \]