# 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.