https://file.aiquickdraw.com/mathbot/file/17268746881452xou6p5d.jpg

The image provides the **Bessel differential equation** of the form:

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

where \( p \) is a constant. This equation is significant in various physical applications, especially in cylindrical coordinate systems, such as in solving problems in transport processes.

### Goal
Given that \( J_p(x) \) is one solution of this equation, the task is to determine the **second independent solution**.

### Method to find the second solution:
The general form of a second-order linear differential equation like this one suggests that the second solution can often be found using the **method of reduction of order**. This method assumes the second solution, \( y_2(x) \), has the form:

\[
y_2(x) = v(x) J_p(x),
\]

where \( v(x) \) is an unknown function to be determined.

After substituting \( y_2(x) \) into the Bessel equation and solving for \( v(x) \), the second solution is typically found to be:

\[
y_2(x) = J_{-p}(x),
\]

for non-integer \( p \), or for integer \( p \), the second solution includes logarithmic terms such as:

\[
y_2(x) = J_p(x) \ln(x) + \text{terms involving } J_p(x).
\]

This gives the complete general solution of the Bessel equation as a linear combination of the two independent solutions.

### Would you like further details on the reduction of order process or clarification on Bessel functions?

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**Relative Questions:**
1. What is the general significance of Bessel functions in physical problems?
2. How does the method of reduction of order help find second solutions to differential equations?
3. What is the importance of the constant \( p \) in Bessel’s equation?
4. How does the solution \( J_p(x) \) behave for small or large values of \( x \)?
5. How are Bessel functions related to other special functions, like Legendre or Hermite functions?

**Tip:** Bessel functions are often used to model wave-like phenomena in cylindrical or spherical systems, such as heat conduction, vibrations, or electromagnetic waves.

Given the second-order differential equation (Bessel's equation): x^2 * d^2y/dx^2 + x * dy/dx + (x^2 - p^2)y = 0, where p is a constant, determine the second solution of the equation, knowing that J_p(x) is one solution.

Explore the Bessel differential equation: x^2 * d^2y/dx^2 + x * dy/dx + (x^2 - p^2)y = 0. Learn how to find the second solution using the reduction of order method, given that J_p(x) is a known solution. This problem applies to advanced calculus and is crucial in cylindrical coordinate systems.

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