Solve for: 2xy * (dy)/(dy)+4x^3=0

Expression: $2xy \times \frac{ \mathrm{d}y }{ \mathrm{d}y }+4{x}^{3}=0$

Move the variable to the right-hand side and change its sign

$2xy \times \frac{ \mathrm{d}y }{ \mathrm{d}y }=-4{x}^{3}$

Separate the differentials

$2xy\mathrm{d}^y=-4{x}^{3}\mathrm{d}^x$

Divide both sides of the equation by $x$

$2y\mathrm{d}^y=\frac{ -4{x}^{3} }{ x }\mathrm{d}^x$

Integrate the left-hand side of the equation with respect to $y$ and the right-hand side of the equation with respect to $x$

$\int{ 2y } \mathrm{d} y=\int{ \frac{ -4{x}^{3} }{ x } } \mathrm{d} x$

Evaluate the integral

${y}^{2}+C_1=\int{ \frac{ -4{x}^{3} }{ x } } \mathrm{d} x$

Evaluate the integral

$\begin{array} { l }{y}^{2}+C_1=-\frac{ 4{x}^{3} }{ 3 }+C_2,& \begin{array} { l }C_1 \in ℝ,& C_2 \in ℝ\end{array}\end{array}$

Since both constants of integration $C_1$ and $C_2$ are arbitrary constants, replace them with the constant $C$

$\begin{array} { l }{y}^{2}=-\frac{ 4{x}^{3} }{ 3 }+C,& C \in ℝ\end{array}$