Solve the following initial value problem,
$$\begin{align}
\frac{dy}{dx} &= \sin(3x)(y + 2)\\
y(\pi/12) &= 2
\end{align}$$This is a separable equation. First we find the general solution following the paradigm.
Separate the variables
$$
\frac{dy}{y + 2} = \sin(3x) dx
$$
Integrate both sides
$$
\log|y + 2| = -(1/3)\cos(3x) + C
$$
Solve for y (if possible)
We simplify the expression to obtain the general solution
$$
y + 2 = k \exp(-(1/3)\cos(3x))
$$
Check for singular solutions
We divided by $y + 2$ which is 0 when $y = -2$, which we can check is a solution. Note this solution is already in the general solution (when the constant is 0).