The solution of the differential equation
`y(2x^(4)+y)(dy)/(dx) = (1-4xy^(2))x^(2)` is given by

Options

A. `3(x^(2)y)^(2)+y^(3)-x^(3)=c`
B. `xy^(2)+y^(3)/3-x^(3)/3+c=0`
C. `2/5yx^(5)+y^(3)/3=x^(3)/3-(4xy^(3))/3+c`
D. None of these

Explanation

To solve the given differential equation:

\[ y(2x^4 + y) \frac{dy}{dx} = (1 - 4xy^2)x^2 \]

we will follow these steps:

### Step 1: Rearranging the Equation
First, we will rearrange the equation by cross-multiplying to isolate \( dy \) and \( dx \).

\[ y(2x^4 + y) dy = (1 - 4xy^2)x^2 dx \]

### Step 2: Expanding and Collecting Terms
Now, we will expand both sides and collect all terms on one side of the equation.

\[ 2x^4 y dy + y^2 dy - x^2 dx + 4x^3 y^2 dx = 0 \]

### Step 3: Factoring
Next, we can factor out common terms from the equation:

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

### Step 4: Identifying Total Derivative
We can recognize that the left-hand side can be expressed as a total derivative. The expression \( d(x^2 y) \) can be derived using the product rule:

\[ d(x^2 y) = 2xy dx + x^2 dy \]

Thus, we rewrite our equation in terms of total derivatives:

\[ 2x^2 y dy + y^2 dy - x^2 dx + 4x^3 y^2 dx = 0 \]

### Step 5: Integrating Both Sides
Now we will integrate both sides. We will separate the variables and integrate:

\[ \int (2x^2 y) dy + \int y^2 dy = \int x^2 dx + \int 4x^3 y^2 dx \]

### Step 6: Performing the Integrations
1. The integral of \( 2x^2 y dy \) is \( x^2 y^2 \).
2. The integral of \( y^2 dy \) is \( \frac{y^3}{3} \).
3. The integral of \( x^2 dx \) is \( \frac{x^3}{3} \).
4. The integral of \( 4x^3 y^2 dx \) is \( 4y^2 \frac{x^4}{4} = y^2 x^4 \).

Putting it all together, we have:

\[ x^2 y^2 + \frac{y^3}{3} - \frac{x^3}{3} = C \]

### Step 7: Rearranging the Equation
Finally, we can rearrange the equation to express it in a standard form:

\[ 3x^2 y^2 + y^3 - x^3 = C' \]

where \( C' = 3C \) is a constant.

### Final Solution
Thus, the solution of the given differential equation is:

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