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 method of separating the variables.

Given differential equation of the form

\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}


By separating the variable technique we must keep the terms dy and dx in the numerators with their respective functions.
Now integration 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}


Which is the required solution of the given differential equation.