Question

Solve the differential equation by separation of variables.

(x + 2) dx = (x + 3) sin y cos y dy

Answer 1

$(x + 2) dx = (x + 3) \sin y. \cos y dy$

$\Rightarrow \frac{x+2}{x+3} dx=\frac{\sin2y}{2}dy$

Integrating both sides, we get

$\int\frac{x+2}{x+3} dx=\int\frac{\sin2y}{2}dy \\\Rightarrow \int\frac{x+3-1}{x+3} dx=\frac{1}{2}\int\sin2ydy \\\Rightarrow \int dx-\int \frac{1}{x+3}dx=-\frac{1}{2}.\frac{\cos 2y}{2}+c \\\Rightarrow x-ln|x+3|=-\frac{\cos 2y}{4}+c$