Solve by separation of variables dy/dx = e^{3x+2y}.

(1) $\begin{equation*} \frac{dy}{dx} = e^{3x+2y} \end{equation*}$

We can separate the variables as follows:

(2) $\begin{equation*} e^{-2y} dy = e^{3x} dx \end{equation*}$

Now, we can integrate both sides of the equation:

(3) $\begin{equation*} \int e^{-2y} dy = \int e^{3x} dx \end{equation*}$

Integrating the left-hand side, we get:

(4) $\begin{equation*} -\frac{1}{2} e^{-2y} = \frac{1}{3} e^{3x} + C_1 \end{equation*}$

where $C_1$ is the constant of integration.

Simplifying and solving for $y$ , we have:

(5) $\begin{equation*} -3e^{-2y} = 2e^{3x} + C \end{equation*}$

where $C = C_1$ is a constant of integration.

Therefore, the solution to the differential equation $\frac{dy}{dx} = e^{3x+2y}$ is:

(6) $\begin{equation*} -3e^{-2y} = 2e^{3x} + C \end{equation*}$

where $C$ is a constant of integration.