We are given the following differential equation:

$x^2 \frac{dy}{dx} = 1 - x^2 + y^2 - x^2y^2$

We aim to solve this differential equation for $y(x)$.

Step 1: Simplify the equation

First, rearrange the given equation:

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

Now, let's express it in a more structured form:

$x^2 \frac{dy}{dx} = (1 - x^2)(1 - y^2)$

Step 2: Separate the variables

Next, separate the variables $y$ and $x$ so that we can integrate:

$\frac{dy}{(1 - y^2)} = \frac{(1 - x^2)}{x^2} dx$

Step 3: Simplify each side

We simplify the right-hand side:

$\frac{1 - x^2}{x^2} = \frac{1}{x^2} - 1$

So now the equation becomes:

$\frac{dy}{(1 - y^2)} = \left( \frac{1}{x^2} - 1 \right) dx$

Step 4: Integrate both sides

We can now integrate both sides separately.

The left-hand side is:

$\int \frac{dy}{1 - y^2} = \tanh^{-1}(y)$

The right-hand side becomes:

$\int \left( \frac{1}{x^2} - 1 \right) dx = -\frac{1}{x} - x$

Step 5: Combine and solve for $y(x)$

Now, we combine the results from the integrals:

$\tanh^{-1}(y) = -\frac{1}{x} - x + C$

Finally, solve for $y(x)$ by applying the hyperbolic tangent:

$y(x) = \tanh\left( -\frac{1}{x} - x + C \right)$

This is the general solution to the differential equation.

Would you like more details on any of the steps? Here are five related questions you could explore next:

1. What are hyperbolic functions, and how are they related to trigonometric functions?
2. How do you handle differential equations that cannot be easily separated?
3. Can you provide examples of differential equations with real-world applications?
4. What is the geometric interpretation of the solution to this equation?
5. How does the integration of hyperbolic functions differ from trigonometric ones?

Tip: When solving differential equations, always look for ways to simplify through variable separation or substitution to make integration easier.