Solve the following initial value problem,
$$\begin{align}
\frac{dy}{dx} &= \frac{y^2 + 5y + 6}{x^2 + 2x}\\
y(3) &= -5
\end{align}$$This is a separable equation. First we find the general solution following the paradigm.
Separate the variables
$$
\frac{dy}{y^2 + 5y + 6} = \frac{dx}{x^2 + 2x}
$$
Integrate both sides
$$
\log|y + 2| - \log|y + 3| = (1/2)\log|x| - (1/2)\log|x + 2| + C
$$
Solve for y (if possible)
We simplify the expression to obtain the general solution
$$
\left(\frac{y + 2}{y + 3}\right)^2 = k \frac{x}{x + 2}
$$
Check for singular solutions
We divided by $y^2 + 5y + 6$ which is 0 when $y = -2$ or $y = -3.$ We can check these are both solutions. Now $y = -2$ is already in the general solution (when the constant is 0), but $y = -3$ is a singular solution.