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To solve the given differential equation

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

with the initial condition \( y(0) = 2 \), we can proceed as follows:

### Step 1: Rewrite the Differential Equation

Rewrite the differential equation in the form:

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

### Step 2: Separate Variables

We can separate the variables \(x\) and \(y\):

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

### Step 3: Integrate Both Sides

Integrate both sides:

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

The integral of \(\frac{1}{1+y^2}\) with respect to \(y\) is \(\tan^{-1}(y)\), and the integral of \(\frac{1}{1+x^2}\) with respect to \(x\) is \(\tan^{-1}(x)\):

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

### Step 4: Solve for the Constant of Integration \(C\)

Use the initial condition \(y(0) = 2\) to find \(C\):

\[
\tan^{-1}(2) = \tan^{-1}(0) + C \Rightarrow C = \tan^{-1}(2)
\]

### Step 5: Write the General Solution

Substituting \(C\) back into the solution:

\[
\tan^{-1}(y) = \tan^{-1}(x) + \tan^{-1}(2)
\]

### Step 6: Solve for \(y\)

Taking the tangent of both sides to express \(y\) explicitly in terms of \(x\):

\[
y = \tan(\tan^{-1}(x) + \tan^{-1}(2))
\]

This is the implicit solution to the differential equation. The form might vary depending on further trigonometric manipulation if desired.

Would you like more details or have any questions?

### Related Questions:

1. How do you handle initial conditions in solving differential equations?
2. What is the significance of separating variables when solving differential equations?
3. What are other methods to solve non-linear differential equations if separation of variables is not possible?
4. How do you interpret the solution \( y = \tan(\tan^{-1}(x) + \tan^{-1}(2)) \) graphically?
5. How does the method change if the initial condition is different?

### Tip

Always verify that the solution satisfies both the differential equation and the initial condition to ensure correctness.

Learn how to solve the differential equation (1 + x^2) dy/dx = 1 + y^2 with the initial condition y(0) = 2. This problem involves key concepts in differential equations, including separation of variables and initial value problems. Suitable for college-level students.

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