Solve the following initial value problem,
$$\begin{align}
\frac{dy}{dx} &= 2\cos(3x)(y^2 - 9y + 20)\\
y(\pi/12) &= 2
\end{align}$$This is a separable equation. First we find the general solution following the paradigm.
Separate the variables
$$
\frac{dy}{y^2 - 9y + 20} = 2\cos(3x) dx
$$
Integrate both sides
$$
\log|y - 5| - \log|y - 4| = (2/3)\sin(3x) + C
$$
Solve for y (if possible)
We simplify the expression to obtain the general solution
$$
\frac{y - 5}{y - 4} = k \exp((2/3)\sin(3x))
$$
Check for singular solutions
We divided by $y^2 - 9y + 20$ which is 0 when $y = 5$ or $y = 4.$ We can check these are both solutions. Now $y = 5$ is already in the general solution (when the constant is 0), but $y = 4$ is a singular solution.