Solve 2x(1-x)y^h+(1-x)y^prime+3y=0

$$2 x ( 1 - x ) y ^ { h } + ( 1 - x ) y ^ { \prime } + 3 y = 0$$

Solve for h (complex solution)

$$\left\{\begin{matrix}h=\log_{y}\left(-\frac{3y}{2x\left(1-x\right)}\right)+\frac{2i\pi n_{1}}{\ln(y)}\text{, }n_{1}\in \mathrm{Z}\text{, }&y\neq 0\text{ and }y\neq 1\text{ and }x\neq 0\text{ and }x\neq 1\\h\in \mathrm{C}\text{, }&y=0\text{ or }\left(x=\frac{\sqrt{7}+1}{2}\text{ and }y=1\right)\text{ or }\left(x=\frac{1-\sqrt{7}}{2}\text{ and }y=1\right)\end{matrix}\right.$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{6y^{h+1}+y^{2h}}}{2y^{h}}+\frac{1}{2}\text{; }x=\frac{\sqrt{6y^{h+1}+y^{2h}}}{2y^{h}}+\frac{1}{2}\text{, }&h=0\text{ or }y\neq 0\\x\in \mathrm{C}\text{, }&y=0\text{ and }h\neq 0\end{matrix}\right.$$

Solve for h

$$\left\{\begin{matrix}h=\log_{y}\left(-\frac{3y}{2x\left(1-x\right)}\right)\text{, }&y>0\text{ and }\left(x<0\text{ or }x>1\right)\text{ and }y\neq 1\\h\in \mathrm{R}\text{, }&\left(y=-1\text{ and }x=\frac{1-\sqrt{7}}{2}\text{ and }Denominator(h)\text{bmod}2=1\text{ and }Numerator(h)\text{bmod}2=1\right)\text{ or }\left(y=-1\text{ and }x=\frac{\sqrt{7}+1}{2}\text{ and }Denominator(h)\text{bmod}2=1\text{ and }Numerator(h)\text{bmod}2=1\right)\text{ or }\left(y=1\text{ and }x=\frac{\sqrt{7}+1}{2}\right)\text{ or }\left(y=1\text{ and }x=\frac{1-\sqrt{7}}{2}\right)\\h>0\text{, }&y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{6y^{h+1}+y^{2h}}}{2y^{h}}+\frac{1}{2}\text{; }x=\frac{\sqrt{6y^{h+1}+y^{2h}}}{2y^{h}}+\frac{1}{2}\text{, }&\left(Denominator(h)\text{bmod}2=1\text{ or }y>0\right)\text{ and }y\neq 0\text{ and }6y^{h+1}+y^{2h}\geq 0\\x\in \mathrm{R}\text{, }&y=0\text{ and }h>0\end{matrix}\right.$$