The solution of differential equation `x^(2)y^(2)dy = (1-xy^(3))dx` is

Options

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

Answer

To solve the differential equation \( x^2 y^2 dy = (1 - xy^3) dx \), we can follow these steps:

### Step 1: Rearranging the Equation
We start by rewriting the equation in a more manageable form:

\[ x^2 y^2 dy = (1 - xy^3) dx \]

Dividing both sides by \( x^2 y^2 \):

\[ dy = \frac{1 - xy^3}{x^2 y^2} dx \]

### Step 2: Separating Variables
Next, we can separate the variables:

\[ y^2 dy = \left(\frac{1}{x^2} - \frac{y^3}{x}\right) dx \]

### Step 3: Integrating Both Sides
Now we integrate both sides. The left side becomes:

\[ \int y^2 dy = \frac{y^3}{3} + C_1 \]

For the right side, we can break it into two parts:

\[ \int \left(\frac{1}{x^2} - \frac{y^3}{x}\right) dx = \int \frac{1}{x^2} dx - \int \frac{y^3}{x} dx \]

Calculating the first integral:

\[ \int \frac{1}{x^2} dx = -\frac{1}{x} + C_2 \]

The second integral involves \( y \), which we treat as a constant during this integration:

\[ -\int \frac{y^3}{x} dx = -y^3 \ln|x| + C_3 \]

### Step 4: Combining Results
Combining the results from both integrals, we have:

\[ \frac{y^3}{3} = -\frac{1}{x} - y^3 \ln|x| + C \]

where \( C \) is a constant that combines \( C_1, C_2, \) and \( C_3 \).

### Step 5: Rearranging the Equation
Rearranging gives us the implicit solution of the differential equation:

\[ \frac{y^3}{3} + y^3 \ln|x| + \frac{1}{x} = C \]

### Final Solution
Thus, the solution to the differential equation is:

\[ \frac{y^3}{3} + y^3 \ln|x| + \frac{1}{x} = C \]