Equations reducible to Bessel's equation

Question

I am solving a differential equation: $$ xy'' + ay' + k^{2}xy = 0, \tag{1} $$ where $a$ and $k$ are some constants, one way to solve this equation is to reduce it to Bessel's form: $$ x^{2}y'' + xy' + (k^{2}x^{2} - n^{2})y = 0, \tag{2} $$ Since a solution is known for Bessel's equation depending upon the nature of $n$. The standard way to reduce $(1)$ to $(2)$ is to make the substitution $$y = x^{n}z,$$where $z = z(x)$. I am interested in knowing how one arrives or how one knows to do that specific substitution to reduce $(1)$ to $(2)$. Can anybody enlighten me or lead me to a source?