We use Bessel $I_v(x)$, Bessel $J_v(x)$, Parabolic Cylinder $D_v(x)$, and Bessel Y Zero $y_{v,x}$
Question:
The goal is to find the simplest parabolic cylinder equation that $y_{\frac34,x}$ solves. A partial solution is here. Please see the motivation section for the $v=\frac14$ solution. However, the first step is to simplify: $$\frac{J_\frac34(x)}{J_{-\frac34}(x)}=\text{Rational expression in terms of }D_v(x) $$ and use algebra to isolate the parabolic cylinder functions as much as possible. Let’s introduce the DLMF Parabolic Cylinder $\text W(a,x)$, it’s derivative wrt $x$ function, and this identity: $$\frac{J_\frac34(x)}{J_{-\frac34}(x)} =\frac{W’(0,2\sqrt x)-W’(0,-2\sqrt x)}{W’(0,2\sqrt x)+W’(0,-2\sqrt x)}$$
Now use this conversion formula and a change of notation to possibly get:
$$\frac2{\frac{J_\frac34\left(\frac{x^2}4\right)}{J_{-\frac34} \left(\frac{x^2}4\right)} +1}-1=\frac{ W’(0,-x)}{W’(0,x)} \mathop=^{\large ?} i(\sqrt2+1)\frac{\frac d{dx}\left(D_{-\frac12}\left(\sqrt[-4]{-1}x\right)-\sqrt[4]{-1} D_{-\frac12}\left(\sqrt[4]{-1}x\right)\right)}{\frac d{dx} \left(D_{-\frac12}\left(\sqrt[-4]{-1}\right)+\sqrt[4]{-1} D_{-\frac12}\left(\sqrt[4]{-1}x\right)\right)}$$
However, expanding the derivatives gives complicated expressions. Is there any way to simplify $\frac{W’(0,-x)}{W’(0,x)}$ or $\frac{J_\frac34(x)}{J_{-\frac34}(x)}$ into a simple ratio of $D_v(x)$?. The following identity may help simplify it:
$$\frac{a-by}{a+by}=\frac2{\frac ba y+1}-1$$
Motivation:
$$\frac{D_{-\frac12}(iz)}{D_{-\frac12}(z)}=\frac{I_{-\frac14}\left(\frac{z^2}4\right)-I_{-\frac14}\left(-\frac{z^2}4\right)}{I_{-\frac14}\left(\frac{z^2}4\right)-I_{-\frac14}\left(\frac{z^2}4\right)},\text{Re}(z)>\text{Im}(z)$$ and an inverse function from
Complex zeroes of Error Function and Parabolic Cylinder Function.
$$\frac{D_{-\frac12}\left(iy\right)}{D_{-\frac12}\left(y\right)}=-e^{2ix}\implies y=(1-i)\sqrt{2y_{\frac14,\Bbb N-\frac{\tan^{-1}(\tan(x))}\pi}}, x\in\Bbb R $$ which works with the $n$th natural number giving the $n$th solution.