Based on the extracted information, let's solve the differential equation:

The differential equation provided is:

\[
(1 + x^2) y'' + x y' - y = 0
\]

with the initial condition:

\[
y(0) = 1
\]

This is a second-order linear homogeneous differential equation, and it resembles a standard form that can be solved using special functions, specifically the **Bessel function**.

### Result of the Solution:

The general solution of the given differential equation is expressed as:

\[
y(x) = C_1 J_0(x) + C_2 Y_0(x)
\]

where:
- \( J_0(x) \) is the Bessel function of the first kind of order 0.
- \( Y_0(x) \) is the Bessel function of the second kind of order 0.

Since you are given the initial condition \( y(0) = 1 \), we use this to determine the constants. However, \( Y_0(x) \) has a logarithmic singularity at \( x = 0 \), meaning \( Y_0(0) \) is undefined. To satisfy the initial condition \( y(0) = 1 \), we eliminate the \( Y_0(x) \) term.

Thus, the solution simplifies to:

\[
y(x) = C_1 J_0(x)
\]

Finally, applying the condition \( y(0) = 1 \), and noting that \( J_0(0) = 1 \), we get \( C_1 = 1 \).

### Final Answer:
\[
y(x) = J_0(x)
\]

This is the solution to the given differential equation.

---

Would you like further details on how the Bessel functions are derived or explanations about their properties?

Here are 5 related questions for further study:
1. How do Bessel functions arise in the context of differential equations?
2. What are the properties of the Bessel function \( J_0(x) \)?
3. Can this type of differential equation be solved using series expansion?
4. What physical problems are modeled by differential equations involving Bessel functions?
5. How do boundary conditions affect the form of the solution?

**Tip:** Bessel functions often appear in problems involving cylindrical or spherical symmetry, so they are common in physics and engineering applications.

Explore the solution to the differential equation (1 + x^2) y'' + x y' - y = 0 with initial condition y(0) = 1. Learn about Bessel functions, their properties, and applications in solving differential equations. Suitable for advanced undergraduate students and beyond.

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