Solution by separation of variables dx+e^{3x}dy= 0.

Given differential equation is:

(1)

We can separate the variables as follows:

(2)

Now, we can integrate both sides of the equation:

(3)

Integrating the left-hand side, we get:

(4)

where is the constant of integration.

For the right-hand side, we can use the substitution and to obtain:

(5)

where is another constant of integration. Substituting back , we have:

(6)

Substituting this back into our original equation, we have:

(7)

where and are constants of integration.

Therefore, the solution to the differential equation is:

(8)

where is the constant of integration.