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.