The differential equation `(dy)/(dx) = (x(1+y^(2)))/(y(1+x^(2))` represents a family of

Options

A. parabola
B. hyperbola
C. circle
D. ellipse

Explanation

To solve the differential equation

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

we will use the method of separation of variables. Here’s a step-by-step solution:

### Step 1: Separate the Variables
We start by rearranging the equation to separate \(y\) and \(x\):

\[ y(1+x^2) \, dy = x(1+y^2) \, dx. \]

### Step 2: Rewrite the Equation
Now, we can rewrite the equation as:

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

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

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

The left side integrates to:

\[ \frac{1}{2} \ln(1+y^2) + C_1, \]

and the right side integrates to:

\[ \frac{1}{2} \ln(1+x^2) + C_2. \]

### Step 4: Combine Constants
We can combine the constants of integration:

\[ \frac{1}{2} \ln(1+y^2) = \frac{1}{2} \ln(1+x^2) + C, \]

where \(C = C_2 - C_1\).

### Step 5: Eliminate the Logarithm
To eliminate the logarithm, we exponentiate both sides:

\[ 1+y^2 = K(1+x^2), \]

where \(K = e^{2C}\).

### Step 6: Rearranging the Equation
Rearranging gives us:

\[ y^2 = K(1+x^2) - 1. \]

### Step 7: Identify the Family of Curves
This can be rewritten as:

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

This represents a family of hyperbolas.

### Conclusion
Thus, the differential equation represents a family of hyperbolas.

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