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For $n=0$, the Bessel's equation simplifies, yielding solutions known as Bessel functions of the first and second kind, $J_0(x)$ and $Y_0(x)$ respectively.

As $x$ approaches infinity, Bessel functions exhibit oscillatory decay, which is characteristic of their asymptotic behavior.

A Bessel's differential equation is of the form

When $n$ is an integer, the general solution is a combination of Bessel functions of the first kind $J_n(x)$ and second kind $Y_n(x)$, which are linearly independent.

Recurrence relations are useful for relating Bessel functions of different orders, and can simplify computation and analysis.

The modified Bessel's equation is like the standard version but with a key difference: $x^2y'' + xy' - (x^2 + n^2)y = 0$; it describes exponentially growing and decaying functions.

For non-integer $n$, solutions involve Bessel functions of the first kind for both $n$ and $-n$, $J_n(x)$ and $J_{-n}(x)$, provided $n$ is not an integer.