Solve the following initial value problem,
$$\begin{align}
\frac{dy}{dx} &= \frac{\exp(-3y)}{2x + 5}\\
y(0) &= 2
\end{align}$$This is a separable equation. First we find the general solution following the paradigm.
Separate the variables
$$
\exp(3y) dy = \frac{dx}{2x + 5}
$$
Integrate both sides
$$
(1/3)\exp(3y) = (1/2)\log|2x + 5| + C
$$
Solve for y (if possible)
We simplify the expression to obtain the general solution
$$
y = (1/3)\log((3/2)\log|2x + 5| + C)
$$
Check for singular solutions
Since we never divided by anything that could be 0, there are no singular solutions.