The problem involves solving the given differential equation:

$(1 + x^2) \, dy + y \, dx = 0, \quad y(1) = 1$

We are provided with multiple-choice answers, and we aim to determine the correct solution. Let us solve this step-by-step.

Step 1: Rewrite the equation

The equation can be rewritten in the following form: $\frac{dy}{y} = -\frac{dx}{1 + x^2}$

This is separable, so we can integrate both sides.

Step 2: Integrate both sides

1. The left-hand side becomes: $\int \frac{1}{y} \, dy = \ln |y| + C_1$

2. The right-hand side is: $\int -\frac{1}{1 + x^2} \, dx = -\arctan(x) + C_2$

Combining: $\ln |y| = -\arctan(x) + C$

Step 3: Solve for $y$

Exponentiate both sides: $y = e^{-\arctan(x) + C} = e^C \cdot e^{-\arctan(x)}$

Let $e^C = C_1$, so: $y = C_1 \cdot e^{-\arctan(x)}$

Step 4: Use the initial condition

The initial condition is $y(1) = 1$. When $x = 1$, we have: $1 = C_1 \cdot e^{-\arctan(1)}$

Recall that $\arctan(1) = \frac{\pi}{4}$, so: $1 = C_1 \cdot e^{-\frac{\pi}{4}}$

Thus: $C_1 = e^{\frac{\pi}{4}}$

Substitute $C_1$ back into the equation: $y = e^{\frac{\pi}{4}} \cdot e^{-\arctan(x)} = e^{\frac{\pi}{4} - \arctan(x)}$

Final Answer:

From the given options, the correct answer is: $\boxed{\text{D) } y = e^{\frac{\pi}{4} - \arctan(x)}}$

Would you like a detailed explanation of any step, or do you have further questions?

Related Questions:

1. What is the process for solving other separable differential equations?
2. How does the initial condition influence the solution to a differential equation?
3. What are the steps to compute $\arctan(x)$ in common angles?
4. Why is it necessary to exponentiate after integrating logarithmic forms?
5. How does the method differ for non-separable differential equations?

Tip:

When solving separable differential equations, always check if the equation can be rewritten into $dy/y = -f(x)dx$ or a similar form to simplify integration.