Answer to Question #237939 in Differential Equations for James

Question

Solve the differential equation by separation of variables with step-by-step procedures.

(x+2)dx = (x+3) siny cosydy

Answer 1

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

$$\sin y \cos ydy=\dfrac{x+2}{x+3}dx$$

Integrate

$$\int \sin y \cos ydy=\int\dfrac{x+2}{x+3}dx$$

$$\int \sin y \cos ydy=\dfrac{1}{2}\int \sin (2y)dy$$

$$=-\dfrac{1}{4}\cos(2y)+C_1$$

$$\int\dfrac{x+2}{x+3}dx=\int\dfrac{x+3}{x+3}dx-\int\dfrac{1}{x+3}dx$$

$$=\int dx-\int\dfrac{1}{x+3}=x-\ln(|x+3|)+C_2$$

$$-\dfrac{1}{4}\cos(2y)+C_1=x-\ln(|x+3|)+C_2$$

The solution of the equation in implicit form is

$$x-\ln(|x+3|)+\dfrac{1}{4}\cos(2y)=C$$