Solve the following initial value problem,
$$\begin{align}
\frac{dy}{dx} &= (x + 3)\exp(-3y)\\
y(0) &= 1
\end{align}$$This is a separable equation. First we find the general solution following the paradigm.
Separate the variables
$$
\exp(3y) dy = (x + 3) dx
$$
Integrate both sides
$$
(1/3)\exp(3y) = (1/2)x^2 + 3x + C
$$
Solve for y (if possible)
We simplify the expression to obtain the general solution
$$
y = (1/3)\log((3/2)x^2 + 9x + C)
$$
Check for singular solutions
Since we never divided by anything that could be 0, there are no singular solutions.