Solve xy^2=4a^2(2a-x)

$$x y ^ { 2 } = 4 a ^ { 2 } ( 2 a - x )$$

Solve for x

$$\left\{\begin{matrix}x=\frac{8a^{3}}{y^{2}+4a^{2}}\text{, }&a\neq 0\text{ or }y\neq 0\\x\in \mathrm{R}\text{, }&a=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 4a^{2} by 2a-x.

$$xy^{2}=8a^{3}-4a^{2}x$$

Add 4a^{2}x to both sides.

$$xy^{2}+4a^{2}x=8a^{3}$$

Combine all terms containing x.

$$\left(y^{2}+4a^{2}\right)x=8a^{3}$$

Divide both sides by y^{2}+4a^{2}.

$$\frac{\left(y^{2}+4a^{2}\right)x}{y^{2}+4a^{2}}=\frac{8a^{3}}{y^{2}+4a^{2}}$$

Dividing by y^{2}+4a^{2} undoes the multiplication by y^{2}+4a^{2}.

$$x=\frac{8a^{3}}{y^{2}+4a^{2}}$$

Solve for a

$$\left\{\begin{matrix}\\a=\frac{2^{\frac{2}{3}}\sqrt[3]{3\sqrt{3}|x||y|\sqrt{4x^{2}+27y^{2}}+27xy^{2}+2x^{3}}}{12}+\frac{2^{\frac{2}{3}}\sqrt[3]{-3|x||y|\sqrt{12x^{2}+81y^{2}}+27xy^{2}+2x^{3}}}{12}+\frac{x}{6}\text{, }&\text{unconditionally}\\a=-\frac{2^{\frac{2}{3}}\sqrt[3]{3\sqrt{3}|x||y|\sqrt{4x^{2}+27y^{2}}+27xy^{2}+2x^{3}}}{12}+\frac{x}{6}\text{, }&\frac{x^{2}y^{4}}{256}+\frac{y^{2}x^{4}}{1728}=0\end{matrix}\right.$$