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.