Solve the following initial value problem,
$$\begin{align}
\frac{dy}{dx} &= \frac{7x^2 + 3x + 4}{y^2 + y + 4}\\
y(0) &= 4
\end{align}$$This is a separable equation. First we find the general solution following the paradigm.
Separate the variables
$$
(y^2 + y + 4) dy = (7x^2 + 3x + 4) dx
$$
Integrate both sides
$$
(1/3)y^3 + (1/2)y^2 + 4y = (7/3)x^3 + (3/2)x^2 + 4x + C
$$
Solve for y (if possible)
In this case, the expression is already about as simple as we can make it.
Check for singular solutions
Since we never divided by anything that could be 0, there are no singular solutions.