I was solving a quantum mechanics problem (harmonic oscillateur) and i need to solve this Weber-Hermite differential equation in an analytic method: $$y"-x^2(y)=0$$ I know the solution of this equation is the parabolic cylinder function but i need to solve it analyticaly (to understand how they get the solution) And thank you!!
2026-03-24 23:59:19.1774396759
Weber-Hermite differential equation
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I do not know if this answers the question.
The general Weber differential equation (the solution of which being $D_\nu (x)$) is $$y''+y \left(\nu +\frac{1}{2}-\frac{t^2}{4}\right)y=0$$ So, for your case, $\nu=-\frac{1}{2}$ and you need to redefine $x=t \sqrt 2$ to get your equation.