Solve the following initial value problem,
$$\begin{align}
\frac{dy}{dx} &= (4x^2 + 8x + 2)\exp(-3y)\\
y(0) &= 2
\end{align}$$This is a separable equation. First we find the general solution following the paradigm.
Separate the variables
$$
\exp(3y) dy = (4x^2 + 8x + 2) dx
$$
Integrate both sides
$$
(1/3)\exp(3y) = (4/3)x^3 + 4x^2 + 2x + C
$$
Solve for y (if possible)
We simplify the expression to obtain the general solution
$$
y = (1/3)\log(4x^3 + 12x^2 + 6x + C)
$$
Check for singular solutions
Since we never divided by anything that could be 0, there are no singular solutions.