`(x^(2)-x^(2)y)(dy)/(dx)+y^(2)+x^(2)y^(2)=0`

Options

A. `x-(x^(-1)+y^(-1))-logy=c`
B. `x+(x^(-1)-y^(-1))=log(cy)`
C. `x^(2)+xy+log((y)/(x))=c`
D. `(x^(-1)-y^(-1))+logy=x+c`

Answer

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

### Step 1: Rearranging the Equation
Start by rearranging the equation to isolate \(\frac{dy}{dx}\):
\[ (x^{2} - x^{2}y) \frac{dy}{dx} = - (y^{2} + x^{2}y^{2}) \]
This can be simplified to:
\[ \frac{dy}{dx} = -\frac{y^{2} + x^{2}y^{2}}{x^{2} - x^{2}y} \]

### Step 2: Factoring Out Common Terms
Factor out common terms in the numerator and denominator:
\[ \frac{dy}{dx} = -\frac{y^{2}(1 + x^{2})}{x^{2}(1 - y)} \]

### Step 3: Separating Variables
Separate the variables \(y\) and \(x\):
\[ \frac{1 - y}{y^{2}} dy = -\frac{1 + x^{2}}{x^{2}} dx \]

### Step 4: Simplifying the Left Side
Rewrite the left side:
\[ \left(\frac{1}{y^{2}} - \frac{1}{y}\right) dy = -\left(\frac{1}{x^{2}} + 1\right) dx \]

### Step 5: Integrating Both Sides
Now, integrate both sides:
\[ \int \left(\frac{1}{y^{2}} - \frac{1}{y}\right) dy = \int \left(-\frac{1}{x^{2}} - 1\right) dx \]

### Step 6: Performing the Integrals
The integrals are computed as follows:
- Left side:
\[ \int \frac{1}{y^{2}} dy = -\frac{1}{y}, \quad \int -\frac{1}{y} dy = -\ln|y| \]
Thus, the left side becomes:
\[ -\frac{1}{y} + \ln|y| \]

- Right side:
\[ \int -\frac{1}{x^{2}} dx = \frac{1}{x}, \quad \int -1 dx = -x \]
Thus, the right side becomes:
\[ -\frac{1}{x} - x \]

### Step 7: Combining the Results
Combine the results of the integrals:
\[ -\frac{1}{y} + \ln|y| = -\frac{1}{x} - x + C \]

### Step 8: Rearranging the Equation
Rearranging gives us:
\[ \ln|y| - \frac{1}{y} + \frac{1}{x} + x = C \]

### Final Result
Thus, the solution to the differential equation is:
\[ \ln|y| - \frac{1}{y} + \frac{1}{x} + x = C \]