Using the indicated substitutions, find a general solution...

Using the indicated substitutions, find a general solution in terms of $J_{r}$ and $Y_{c^{\prime}}$. Indicate whether you could also use $J_{-v}$ instead of $Y_{r}$. (Show the details of your work.)
$$x y^{\prime \prime}+11 y^{\prime}+x y=0 \quad\left(y=x^{-5} u\right)$$

Key Concepts

Bessel Functions of the Second Kind (Y_v)

Also known as Neumann functions, these form the second set of independent solutions to Bessel’s differential equation, particularly when a second linearly independent solution is needed. They typically exhibit a singularity at the origin and are crucial for constructing the general solution when combined with J_v.

Bessel Functions of the First Kind (J_v)

These functions constitute one set of fundamental solutions to Bessel’s differential equation. They are regular at the origin for non-negative orders and often used to express the general solution to problems where boundary conditions require the solution to be finite as the independent variable approaches zero.

Linear Independence and Alternative Representations

For a complete general solution of a linear differential equation, two linearly independent solutions are required. In the Bessel context, when the order is not an integer, J_v and J_{-v} can serve as independent solutions. However, in cases where J_v and J_{-v} are not independent (for instance, if the order is an integer), the use of Y_v is necessary. Thus, specifying whether J_{-v} can be used instead of Y_v depends on the order of the Bessel function being non-integer.

Bessel's Differential Equation

This is a second?order linear differential equation whose solutions are the Bessel functions. It arises when using a substitution to transform a variable-coefficient ordinary differential equation, which in many physical problems (like cylindrical symmetry problems) naturally leads to an equation that matches the standard form of Bessel’s differential equation.

Substitution Techniques in Differential Equations

Substitution methods involve changing variables in order to simplify a differential equation or transform it into a well-known form. In the context of the given problem, a specific substitution (here y = x^(-5) u) is used to recast the original equation into one that can be identified as Bessel’s equation, thereby making it possible to employ known solutions in terms of Bessel functions.

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