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

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

has a particular solution:
 * $y = \dfrac {\sin x} {\sqrt x}$

Proof
Note that:

Inserting into $(1)$, and gradually simplifying:

hence demonstrating that:
 * $y_1 = \dfrac {\sin x} {\sqrt x}$

is a particular solution of $(1)$.