Question

Solve (1-4xy^2)(dy/dx)=y^3 by regarding y as the independent variable rather than x?????

Answer 1

\[\begin{align*}
(1-4xy^2)\frac{dy}{dx}&=y^3\\\\
\frac{dy}{dx}&=\frac{y^3}{1-4xy^2}\\\\
\frac{dx}{dy}&=\frac{1-4xy^2}{y^3}\\\\
\frac{dx}{dy}&=\frac{1}{y^3}-\frac{4x}{y}
\end{align*}\]
Now you have an ODE linear in \(x\).