# Solve the Differential Equation y’=y^2 Sinx

In this tutorial we shall solve a differential equation of the form $y’ = {y^2}\sin x$ by using the separating the variables method.

The differential equation of the form is given as
$y’ = {y^2}\sin x$

This differential equation can also be written as
$\frac{{dy}}{{dx}} = {y^2}\sin x$

Separating the variables, the given differential equation can be written as
$\begin{gathered} \frac{1}{{{y^2}}}dy = \sin xdx \\ \Rightarrow {y^{ – 2}}dy = \sin xdx\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Keep in mind that in the separating variable technique the terms $dy$ and $dx$ are placed in the numerator with their respective variables.

Now integrating both sides of the equation (i), we have
$\int {{y^{ – 2}}dy = \int {\sin xdx} }$

Using the formulas of integration $\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}}}$ and $\int {\sin xdx = – \cos x}$, we get
$\begin{gathered} \frac{{{y^{ – 2 + 1}}}}{{ – 2 + 1}} = – \cos x + c \\ \Rightarrow \frac{{{y^{ – 1}}}}{{ – 1}} = – \cos x + c \\ \Rightarrow – \frac{1}{y} = – \cos x + c \\ \Rightarrow cy – y\cos x + 1 = 0 \\ \end{gathered}$

This is the required solution of the given differential equation.