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

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

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

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

which is in the form:
 * $y'' + P \left({x}\right) y' + Q \left({x}\right) y = 0$

where:
 * $P \left({x}\right) = \dfrac 1 x$
 * $Q \left({x}\right) = 1 - \dfrac 1 {4 x^2}$

From Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another:
 * $y_2 \left({x}\right) = v \left({x}\right) y_1 \left({x}\right)$

where:
 * $\displaystyle v = \int \dfrac 1 { {y_1}^2} e^{-\int P \, \mathrm d x} \, \mathrm d x$

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:


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