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

Options

A. `x+y=log(cxy)`
B. `log((x)/(y))=(1)/(x)-(1)/(y)+c`
C. `log((y)/(x))=(1)/(x)-(1)/(y)+c`
D. `log((x)/(y))=(1)/(x)+(1)/(y)+c`

Answer

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

### Step 1: Rearranging the Equation
We start by rewriting the given equation:
\[ (x^2 - yx^2)dy + (y^2 + xy^2)dx = 0 \]
This can be rearranged to:
\[ (x^2(1 - y))dy = -(y^2(1 + x))dx \]

### Step 2: Separating Variables
Now, we can separate the variables \(y\) and \(x\):
\[ \frac{1 - y}{y^2} dy = -\frac{1 + x}{x^2} dx \]

### Step 3: Integrating Both Sides
Next, we integrate both sides:
\[ \int \frac{1 - y}{y^2} dy = -\int \frac{1 + x}{x^2} dx \]

#### Left Side Integration
For the left side:
\[ \int \frac{1 - y}{y^2} dy = \int \left( y^{-2} - y^{-1} \right) dy = \left( -\frac{1}{y} - \log |y| \right) \]

#### Right Side Integration
For the right side:
\[ -\int \frac{1 + x}{x^2} dx = -\left( \int x^{-2} dx + \int x^{-1} dx \right) = -\left( -\frac{1}{x} + \log |x| \right) = \frac{1}{x} - \log |x| \]

### Step 4: Combining Results
Now we combine the results of the integrations:
\[ -\frac{1}{y} - \log |y| = \frac{1}{x} - \log |x| + C \]
Where \(C\) is the constant of integration.

### Step 5: Rearranging the Equation
Rearranging gives us:
\[ -\frac{1}{y} - \frac{1}{x} = \log |x| - \log |y| + C \]
This can be rewritten as:
\[ -\left( \frac{1}{y} + \frac{1}{x} \right) = \log \left| \frac{x}{y} \right| + C \]

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