Solve cosec(y)dx+sec^2(x)dy=0 by separable method

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Given, the differential equation is

(1)   \begin{equation*}cosec(y)dx+sex^2(x)dy=0\end{equation*}

We have to solve the equation by separation of variables.

We can rewrite the above equation as

(2)   \begin{equation*}cosec(y)dx+sex^2(x)dy=0\end{equation*}

We know, cosec y = \frac{1}{siny} and sec x = \frac{1}{cosx}

(3)   \begin{equation*}\frac{1}{sin{y}}dx=-\frac{1}{cos^2{x}}dy\end{equation*}

To solve this differential equation, we can separate the variables by putting all the y terms on one side and all the x terms on the other side:

(4)   \begin{equation*}cos^2{x}dx=-sin{y}dy\end{equation*}

Now we can integrate both sides with respect to their respective variables:

(5)   \begin{equation*}\int cos^2{x}dx= \int -sin{y}dy\end{equation*}

We know that, cos^2{y} = \frac{1+cos(2x)}{2}

We can rewrite the above equation as

(6)   \begin{equation*}\int \frac{1+cos(2x)}{2}dx= \int -sin{y}dy\end{equation*}

Now we can integrate both sides with respect to their respective variables:

On the left side, we get;

(7)   \begin{equation*}\int \frac{1+cos(2x)}{2}dx=\frac{x}{2}+\frac{sin(2x)}{4}+C_1\end{equation*}

On the right side, we get;

(8)   \begin{equation*} \int -sin{y}dy=cosy\end{equation*}

We can simplify this expression as

(9)   \begin{equation*}\frac{x}{2}+\frac{sin(2x)}{4}+C_1=cosy\end{equation*}

By multiplying 4, we get this expression as

(10)   \begin{equation*}2x+{sin(2x)}+C=4cosy\end{equation*}

Hence we can Solve cosec(y)dx+sec^2(x)dy=0 by separable method easily.

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