If `y=f(x)` is solution of differentiable equation `(dy)/(dx)=y/x((1-3x^(2)y^(3)))/((2x^(2)y^(3)+1))`, then (`f(x).x)^(3)` is equal to

Options

A. `x+cy`
B. `2x+cy`
C. `x^(2)+cy`
D. `x^(3)+cy^(2)`

Explanation

To solve the given differential equation and find the value of \((f(x) \cdot x)^3\), we will follow these steps:

### Step 1: Write down the differential equation
We start with the given differential equation:
\[ \frac{dy}{dx} = \frac{y}{x} \cdot \frac{1 - 3x^2 y^3}{2x^2 y^3 + 1} \]

### Step 2: Cross-multiply to rearrange the equation
We can rearrange the equation by cross-multiplying:
\[ (2x^2 y^3 + 1) dy = y(1 - 3x^2 y^3) \frac{dx}{x} \]

### Step 3: Expand and rearrange the terms
Expanding both sides, we get:
\[ 2x^2 y^3 dy + dy = y \left( \frac{dx}{x} - 3x^2 y^3 \frac{dx}{x} \right) \]
This simplifies to:
\[ 2x^2 y^3 dy + dy = y \frac{dx}{x} - 3x^2 y^4 \frac{dx}{x} \]

### Step 4: Collect terms involving \(dy\) and \(dx\)
Rearranging gives us:
\[ 2x^2 y^3 dy + 3x^2 y^4 \frac{dx}{x} = y \frac{dx}{x} - dy \]
This can be rearranged to:
\[ 2x^2 y^3 dy + 3x^2 y^4 \frac{dx}{x} + dy = y \frac{dx}{x} \]

### Step 5: Factor out common terms
Factoring out common terms leads us to:
\[ (2x^2 y^3 + 3x^2 y^4) dy = y \frac{dx}{x} - dy \]

### Step 6: Separate variables
We can separate the variables:
\[ \frac{dy}{y} = \frac{dx}{x(2x^2 y^3 + 3x^2 y^4)} \]

### Step 7: Integrate both sides
Integrating both sides gives:
\[ \int \frac{dy}{y} = \int \frac{dx}{x(2x^2 y^3 + 3x^2 y^4)} \]

### Step 8: Solve the integrals
The left side integrates to:
\[ \ln |y| = \ln |x| + C \]
Exponentiating both sides, we have:
\[ y = kx \quad \text{(where \(k = e^C\))} \]

### Step 9: Substitute back to find \(f(x)\)
Substituting \(y = f(x) = kx\) into our equation gives:
\[ f(x) \cdot x = kx^2 \]

### Step 10: Find \((f(x) \cdot x)^3\)
Now we compute:
\[ (f(x) \cdot x)^3 = (kx^2)^3 = k^3 x^6 \]

### Final Result
Thus, we have:
\[ (f(x) \cdot x)^3 = k^3 x^6 \]
And since \(k\) is a constant that can be absorbed into the constant of integration, we can express the final answer as:
\[ x^6 + C \]