Question

Reduce each of the following differential equations to the variables separable form and hence solve.
(i) $1+\frac{dy}{dx}=\csc (x+y)$

Answer 1

$1+\frac{dy}{dx}=cosec(x+y)$

Let $x+y=t$

$\Rightarrow 1+\frac{dy}{dx}=\frac{dt}{dx}$

So, $\frac{dt}{dx}=cosect$

$\Rightarrow \int \sin tdt=\int dx$

$\Rightarrow -\cos t+c=x$

$\therefore x+\cos (x+y)=c$.