Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0

Theorem
The special case of Bessel's equation:
 * $(1): \quad x^2 y'' + x y' + \paren {x^2 - \dfrac 1 4} y = 0$

has the general solution:
 * $y = C_1 \dfrac {\sin x} {\sqrt x} + C_2 \dfrac {\cos x} {\sqrt x}$

Particular Solution
$(1)$ can be expressed as:
 * $(2): \quad y'' + \dfrac 1 x y' + \paren {1 - \dfrac 1 {4 x^2} } y = 0$

which is in the form:
 * $y'' + \map P x y' + \map Q x y = 0$

where:
 * $\map P x = \dfrac 1 x$
 * $\map Q x = 1 - \dfrac 1 {4 x^2}$

From Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another:
 * $\map {y_2} x = \map v x \, \map {y_1} x$

where:
 * $\ds v = \int \dfrac 1 { {y_1}^2} e^{-\int P \rd x} \rd x$

is also a particular solution of $(1)$.

We have that:

Hence:

and so:

From Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solution:


 * $y = C_1 \dfrac {\sin x} {\sqrt x} + C_2 \dfrac {\cos x} {\sqrt x}$