# Solve the Differential Equation dy/dx=xy^2

In this tutorial we shall evaluate the simple differential equation of the form $$\frac{{dy}}{{dx}} = x{y^2}$$ by using the method of separating the variables.

The differential equation of the form is given as

\[\frac{{dy}}{{dx}} = x{y^2}\]

Separating the variables, the given differential equation can be written as

\[\begin{gathered} \frac{1}{{{y^2}}}dy = xdx \\ \Rightarrow {y^{ – 2}}dy = xdx\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered} \]

In the separating the variables technique we must keep the terms $$dy$$ and $$dx$$ in the numerators with their respective functions.

Now integrating both sides of the equation (i), we have

\[\int {{y^{ – 2}}dy = \int {xdx} } \]

Using the formulas of integration $$\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}}} $$, we get

\[\begin{gathered} \frac{{{y^{ – 2 + 1}}}}{{ – 2 + 1}} = \frac{1}{2}{x^2} + c \\ \Rightarrow \frac{{{y^{ – y}}}}{{ – 1}} = \frac{1}{2}{x^2} + c \\ \Rightarrow – \frac{1}{y} = \frac{1}{2}{x^2} + c \\ \Rightarrow \frac{1}{2}{x^2}y + cy + 1 = 0 \\ \end{gathered} \]

This is the required solution of the given differential equation.