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The provided equation is a second-order differential equation known as **Bessel's equation**, specifically:

\[
x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - p^2\right)y = 0
\]

This is the standard form of Bessel's equation, where \( p \) is a constant. The Bessel equation frequently appears in problems involving cylindrical coordinates, such as heat conduction or wave propagation.

One known solution to this equation is \( J_p(x) \), the Bessel function of the first kind. You are asked to determine the second solution of the equation.

### Steps to find the second solution:

1. **First Solution:**
   - \( y_1(x) = J_p(x) \) is a known solution. The second solution can typically be found using a method called *reduction of order*, which assumes that the second solution is of the form:
     \[
     y_2(x) = v(x) J_p(x)
     \]
     where \( v(x) \) is an unknown function to be determined.

2. **Reduction of Order:**
   To find \( v(x) \), substitute \( y_2(x) = v(x) J_p(x) \) into the original Bessel equation. After simplifying, this will lead to a differential equation for \( v(x) \). This method typically results in the second solution being a Bessel function of the second kind, denoted by \( Y_p(x) \).

3. **Second Solution:**
   - The second linearly independent solution to Bessel's equation is given by \( Y_p(x) \), the Bessel function of the second kind. Therefore, the general solution to the Bessel equation is a linear combination of both solutions:
     \[
     y(x) = A J_p(x) + B Y_p(x)
     \]
     where \( A \) and \( B \) are constants determined by boundary conditions.

Thus, the second solution to the Bessel equation is \( Y_p(x) \).

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Would you like further details on how to derive this using the method of reduction of order or more context on Bessel functions?

Here are some related questions you might find interesting:
1. How do Bessel functions of the first and second kind differ?
2. What is the physical significance of the second solution \( Y_p(x) \) in cylindrical coordinate problems?
3. Can the method of reduction of order be applied to other types of differential equations?
4. How are boundary conditions used to determine the constants in the general solution?
5. What are modified Bessel functions, and how are they related to the standard Bessel functions?

**Tip:** Bessel functions often appear in boundary value problems involving cylindrical or spherical symmetry, making them crucial in engineering and physics.

Determine the second solution of the Bessel equation: x^2 d²y/dx² + x dy/dx + (x² - p²)y = 0.

Explore how to determine the second solution to the Bessel equation: x² d²y/dx² + x dy/dx + (x² - p²)y = 0. Learn about the Bessel functions of the first and second kinds and how they solve differential equations in cylindrical coordinates. Suitable for university-level students.

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