Solve the following initial value problem,
$$\begin{align}
\frac{dy}{dx} &= 7\cos(2x)(y + 5)\\
y(\pi/12) &= 4
\end{align}$$This is a separable equation. First we find the general solution following the paradigm.
Separate the variables
$$
\frac{dy}{y + 5} = 7\cos(2x) dx
$$
Integrate both sides
$$
\log|y + 5| = (7/2)\sin(2x) + C
$$
Solve for y (if possible)
We simplify the expression to obtain the general solution
$$
y + 5 = k \exp((7/2)\sin(2x))
$$
Check for singular solutions
We divided by $y + 5$ which is 0 when $y = -5$, which we can check is a solution. Note this solution is already in the general solution (when the constant is 0).