Solve the differential equation
`(x^(2)-x^(2)y)dy+(y^(2)+xy^(2))dx=0`

Options

A. `log(xy)=x+y+c`
B. `log((x)/(y))=x+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

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

we will follow these steps:

### Step 1: Rearrange the equation
We can rearrange the given equation as:

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

### Step 2: Separate variables
Now, we can separate the variables by dividing both sides by \(x^2y^2\):

\[ \frac{dy}{y^2} + \frac{dx}{x^2} = 0. \]

### Step 3: Integrate both sides
Next, we integrate both sides:

\[ \int \frac{dy}{y^2} + \int \frac{dx}{x^2} = 0. \]

The integrals yield:

\[ -\frac{1}{y} - \frac{1}{x} = C, \]

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

### Step 4: Rearranging the equation
We can rearrange this equation to express it in a more standard form:

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

### Step 5: Final solution
This can be rewritten as:

\[ \frac{1}{x} + \frac{1}{y} = k, \]

where \(k = -C\).

Thus, the general solution to the differential equation is:

\[ \frac{1}{x} + \frac{1}{y} = k. \]