Question

𝑦``+𝑥𝑦`+(𝑥2+2)𝑦=0

Answer 1

Given the equation

$$y''+xy'+(x^2+2)y=0$$

The above is a Sturm-Liouville equation which can be written as:

$$\frac{d}{d x}\left(e^{x^{2} / 2} y^{\prime}(x)\right)+e^{x^{2} / 2}\left(2+x^{2}\right) y(x)=0$$

The solution of the above equation is:

$\begin{gathered} y(x)=c_{1} e^{-1 / 4 i(\sqrt{3}+-i) x^{2}} H_{-\frac{1}{2}} i(-i+\sqrt{3})\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{3} x\right)+ \\ c_{2} e^{-1 / 4 i(\sqrt{3}+-i) x^{2}}{ }_{1} F_{1}\left(\frac{1}{4}+\frac{i \sqrt{3}}{4} ; \frac{1}{2} ; \frac{1}{2} i \sqrt{3} x^{2}\right) \end{gathered}$

Where

$H_n(x)$ is the nth polynomial in x; and

$_1F_1(a;b;x)$ is the Kummer confluent hypergeometric function.