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The given differential equation is:

\[
(1 + x^2) \, dy + (1 + y^2) \, dx = 0
\]

We can solve this equation by using the method of **separation of variables**. Here's the step-by-step process:

### Step 1: Rearrange the equation to separate variables

We aim to express all terms involving \(y\) on one side and all terms involving \(x\) on the other side.

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

Now, divide both sides by \((1 + y^2)\) and \((1 + x^2)\):

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

### Step 2: Integrate both sides

Now we integrate both sides separately.

- The integral on the left-hand side:
\[
\int \frac{1}{1 + y^2} \, dy = \arctan(y)
\]

- The integral on the right-hand side:
\[
\int \frac{-1}{1 + x^2} \, dx = -\arctan(x)
\]

### Step 3: Combine the results

After integrating, we get:

\[
\arctan(y) = -\arctan(x) + C
\]

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

### Step 4: Solve for \(y\)

To express \(y\) in terms of \(x\), take the tangent of both sides:

\[
y = \tan\left(-\arctan(x) + C\right)
\]

Using the tangent addition formula, this simplifies to:

\[
y = \frac{\tan(C) - x}{1 + x\tan(C)}
\]

Thus, the general solution to the differential equation is:

\[
y = \frac{A - x}{1 + xA}
\]

where \(A = \tan(C)\) is a constant.

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Would you like more details or have any questions?

### Related Questions:
1. How can we verify the solution of a separable differential equation?
2. What other methods can be used to solve first-order differential equations?
3. How does the tangent addition formula simplify the solution in this case?
4. What are common applications of separable differential equations?
5. How do the constants of integration affect the family of solutions?

### Tip:
When solving separable differential equations, always double-check whether the equation is already separable or can be rearranged into a separable form before integrating.

Learn how to solve the separable differential equation (1 + x^2) dy + (1 + y^2) dx = 0 using the method of separation of variables. This solution covers key calculus concepts like integration and the tangent addition formula. Suitable for undergraduate students studying calculus.

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