Why Are Parabolic Cylinder Functions Standard Solutions to the Weber Equation?

[intervoxel]

Abramovitz presents even and odd solutions to the Weber equation.

He also presents standard solutions as a pair of parabolic cylinder functions.
Clearly any linear combination of the even and odd solutions is also a solution of the equation.

My question is: Why is the parabolic cylinder function so special to be considered a "standard" solution?

 

[Petar Mali]

Abramovitz presents even and odd solutions to the Weber equation.

He also presents standard solutions as a pair of parabolic cylinder functions.
Clearly any linear combination of the even and odd solutions is also a solution of the equation.

My question is: Why is the parabolic cylinder function so special to be considered a "standard" solution?

Can you write this solution?

 

[intervoxel]

Weber equation
$$\frac{d^2y}{dx^2}-(x^2/4+a)y=0$$

Even solution
$$y_1=e^{-x^2/2}M(\frac{a}{2}+\frac{1}{4},\frac{1}{2},\frac{x^2}{2})$$

Odd solution
$$y_2=xe^{x^2/2}M(-\frac{a}{2}+\frac{1}{4},\frac{1}{2},-\frac{x^2}{2})$$

where M is the Kummer function.

Independent parabolic cylinder functions

$$D_\nu(x)$$ and $$D_{-\nu-1}(ix)$$