Solve x^2y^n+5xy^2+4y=0

$$x ^ { 2 } y ^ { n } + 5 x y ^ { 2 } + 4 y = 0$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=\log_{y}\left(-\frac{y\left(5xy+4\right)}{x^{2}}\right)+\frac{2\pi n_{1}i}{\ln(y)}\text{, }n_{1}\in \mathrm{Z}\text{, }&y\neq 0\text{ and }x\neq -\frac{4}{5y}\text{ and }y\neq 1\text{ and }x\neq 0\\n\in \mathrm{C}\text{, }&\left(x=-4\text{ and }y=1\right)\text{ or }\left(x=-1\text{ and }y=1\right)\text{ or }y=0\end{matrix}\right.$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{y\left(25y^{3}-16y^{n}\right)}-5y^{2}}{2y^{n}}\text{; }x=-\frac{\sqrt{y\left(25y^{3}-16y^{n}\right)}+5y^{2}}{2y^{n}}\text{, }&n=0\text{ or }y\neq 0\\x\in \mathrm{C}\text{, }&y=0\text{ and }n\neq 0\end{matrix}\right.$$

Solve for n

$$\left\{\begin{matrix}n=\log_{y}\left(-\frac{y\left(5xy+4\right)}{x^{2}}\right)\text{, }&x<-\frac{4}{5y}\text{ and }x\neq 0\text{ and }y\neq 1\text{ and }y>0\\n\in \mathrm{R}\text{, }&\left(y=-1\text{ and }x=4\text{ and }Denominator(n)\text{bmod}2=1\text{ and }Numerator(n)\text{bmod}2=1\right)\text{ or }\left(y=-1\text{ and }x=1\text{ and }Denominator(n)\text{bmod}2=1\text{ and }Numerator(n)\text{bmod}2=1\right)\text{ or }\left(y=1\text{ and }x=-4\right)\text{ or }\left(y=1\text{ and }x=-1\right)\\n>0\text{, }&y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{y\left(25y^{3}-16y^{n}\right)}-5y^{2}}{2y^{n}}\text{; }x=-\frac{\sqrt{y\left(25y^{3}-16y^{n}\right)}+5y^{2}}{2y^{n}}\text{, }&\left(Denominator(n)\text{bmod}2=1\text{ or }y>0\right)\text{ and }y\neq 0\text{ and }25y^{4}-16y^{n+1}\geq 0\\x\in \mathrm{R}\text{, }&y=0\text{ and }n>0\end{matrix}\right.$$