Solve the following initial value problem,
$$\begin{align}
\frac{dy}{dx} &= \frac{y + 4}{x^2 + 3x + 2}\\
y(1) &= 4
\end{align}$$This is a separable equation. First we find the general solution following the paradigm.
Separate the variables
$$
\frac{dy}{y + 4} = \frac{dx}{x^2 + 3x + 2}
$$
Integrate both sides
$$
\log|y + 4| = \log|x + 1| - \log|x + 2| + C
$$
Solve for y (if possible)
We simplify the expression to obtain the general solution
$$
y + 4 = k \frac{x + 1}{x + 2}
$$
Check for singular solutions
We divided by $y + 4$ which is 0 when $y = -4$, which we can check is a solution. Note this solution is already in the general solution (when the constant is 0).