Solve by separation of variables dy/dx = 2xy^2.

Given differential equation is:

(1) $\begin{equation*} \frac{dy}{dx} = 2xy^2 \end{equation*}$

We can separate the variables as follows:

(2) $\begin{equation*} \frac{1}{y^2} dy = 2x dx \end{equation*}$

Now, we can integrate both sides of the equation:

(3) $\begin{equation*} \int \frac{1}{y^2} dy = \int 2x dx \end{equation*}$

Integrating the left-hand side, we get:

(4) $\begin{equation*} -\frac{1}{y} = x^2 + C_1 \end{equation*}$

where $C_1$ is the constant of integration.

Simplifying and solving for $y$ , we have:

(5) $\begin{equation*} y = -\frac{1}{x^2 + C} \end{equation*}$

where $C = -\frac{1}{C_1}$ is a constant of integration.

Therefore, the solution to the differential equation $\frac{dy}{dx} = 2xy^2$ is:

(6) $\begin{equation*} y = -\frac{1}{x^2 + C} \end{equation*}$

where $C$ is a constant of integration.