The solution of the differential eqaution
`(x^(2)-yx^(2))(dy)/(dx)+y^(2)+xy^(2)=0`, is

Options

A. `log((x)/(y))=(1)/(x)+(1)/(y)+C`
B. `log((y)/(x))=(1)/(x)+(1)/(y)+C`
C. `log(xy)=(1)/(x)+(1)/(y)+C`
D. `log(xy)+(1)/(x)+(1)/(y)=C`

Answer

To solve the differential equation

\[ (x^2 - yx^2) \frac{dy}{dx} + (y^2 + xy^2) = 0, \]

we can follow these steps:

### Step 1: Rearranging the Equation
First, we rearrange the equation to isolate \(\frac{dy}{dx}\):

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

### Step 2: Dividing by \(x^2 - yx^2\)
Next, we divide both sides by \(x^2 - yx^2\):

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

### Step 3: Factoring the Right Side
Now, we can factor the right side:

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

### Step 4: Separating Variables
We can separate the variables \(y\) and \(x\):

\[ \frac{1 - y}{y^2} dy = -\frac{1 + x}{x^2} dx. \]

### Step 5: Integrating Both Sides
Now we integrate both sides:

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

The left side integrates to:

\[ -\frac{1}{y} - \log |y|, \]

and the right side integrates to:

\[ \frac{1}{x} + \log |x| + C, \]

where \(C\) is the constant of integration.

### Step 6: Combining Results
Combining the results from both integrations, we have:

\[ -\frac{1}{y} - \log |y| = -\frac{1}{x} - \log |x| + C. \]

### Step 7: Rearranging the Equation
Rearranging gives us:

\[ \log |y| + \frac{1}{y} = \log |x| + \frac{1}{x} + C. \]

This is the implicit solution to the differential equation.

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

\[ \log |y| + \frac{1}{y} = \log |x| + \frac{1}{x} + C. \]

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