The solutio of the differential equation `(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`

Explanation

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 can rearrange the equation by factoring out common terms. We can factor out \(x^2\) from the first term and \(y^2\) from the second term:

\[ x^2(1 - y) \frac{dy}{dx} + y^2(1 + x) = 0. \]

### Step 2: Isolate \(\frac{dy}{dx}\)
Next, we can isolate \(\frac{dy}{dx}\):

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

Dividing both sides by \(x^2(1 - y)\) gives:

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

### Step 3: Separate Variables
Now we separate the variables \(y\) and \(x\):

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

### Step 4: Integrate Both Sides
Now we integrate both sides. The left side becomes:

\[ \int \left( \frac{1}{y^2} - \frac{1}{y} \right) dy = \int \left( \frac{1}{y^2} dy - \frac{1}{y} dy \right), \]

which integrates to:

\[ -\frac{1}{y} + \ln |y| + C_1. \]

The right side becomes:

\[ \int -\left( \frac{1}{x^2} + \frac{1}{x} \right) dx = -\left( -\frac{1}{x} + \ln |x| \right) + C_2, \]

which simplifies to:

\[ \frac{1}{x} - \ln |x| + C_2. \]

### Step 5: Combine the Results
Setting the two integrals equal to each other gives:

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

### Step 6: Rearranging the Equation
Rearranging this equation leads to:

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

### Final Step: Exponentiation
Exponentiating both sides gives us the solution in implicit form:

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

This is the general solution to the given differential equation.

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