Solve the following initial value problem,
$$\begin{align}
\frac{dy}{dx} &= \frac{y^2 - 9y + 20}{2x + 4}\\
y(0) &= 2
\end{align}$$This is a separable equation. First we find the general solution following the paradigm.
Separate the variables
$$
\frac{dy}{y^2 - 9y + 20} = \frac{dx}{2x + 4}
$$
Integrate both sides
$$
\log|y - 5| - \log|y - 4| = (1/2)\log|2x + 4| + C
$$
Solve for y (if possible)
We simplify the expression to obtain the general solution
$$
\left(\frac{y - 5}{y - 4}\right)^2 = k (2x + 4)
$$
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.