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

Given differential equation is:

(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.

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