To find the solution to the given differential equation:
[tex]\[
3 y^3 e^{3 x y} - 1 + (2 y + 3 x y^2) e^{3 x y} y^{\prime} = 0,
\][/tex]
we will solve it step-by-step.
### Step 1: Identifying the structure of the equation.
First, we rewrite the equation for clarity:
[tex]\[
3 y^3 e^{3 x y} - 1 + \left(2 y + 3 x y^2 \right) e^{3 x y} y^{\prime} = 0.
\][/tex]
### Step 2: Simplify and factor the equation.
Observe that the term [tex]\( (2 y + 3 x y^2) e^{3 x y} \)[/tex] can be factored out from the left-hand side of the equation, leading us to:
[tex]\[
3 y^3 e^{3 x y} - 1 + e^{3 x y} \left(2 y + 3 x y^2 \right) y^{\prime} = 0.
\][/tex]
### Step 3: Isolate [tex]\( y^{\prime} \)[/tex].
We will move the constant term [tex]\(-1\)[/tex] to the right-hand side:
[tex]\[
3 y^3 e^{3 x y} + \left(2 y + 3 x y^2 \right) e^{3 x y} y^{\prime} = 1.
\][/tex]
Now, distribute [tex]\( e^{3 x y} \)[/tex] across the terms in the parentheses:
[tex]\[
3 y^3 e^{3 x y} + (2 y e^{3 x y} + 3 x y^2 e^{3 x y}) y^{\prime} = 1.
\][/tex]
### Step 4: Combine like terms.
Notice that each term contains [tex]\( e^{3 x y} \)[/tex]. Factor [tex]\( e^{3 x y} \)[/tex] out from the entire left side:
[tex]\[
e^{3 x y} (3 y^3 + (2 y + 3 x y^2) y^{\prime}) = 1.
\][/tex]
### Step 5: Divide by [tex]\( e^{3 x y} \)[/tex]:
We divide both sides of the equation to remove the exponential term:
[tex]\[
3 y^3 + (2 y + 3 x y^2) y^{\prime} = \frac{1}{e^{3 x y}}.
\][/tex]
### Step 6: Integrate terms.
Now we solve it by separating variables and integrating. The restructuring and simplification suggest an integrated function involving [tex]\(y\)[/tex] and [tex]\(x\)[/tex]. Given the possible choices, we can test each given solution:
[tex]\[
\text{a. } y^2 e^{3 x y} - x e^{3 x y} + x + c = 0,
\][/tex]
[tex]\[
\text{b. } y^2 e^{3 x y} - x + c = 0,
\][/tex]
[tex]\[
\text{c. } y^3 e^{3 x y} - x e^{3 x y} + x + c = 0,
\][/tex]
[tex]\[
\text{d. None of the listed solutions.}
\][/tex]
[tex]\[
\text{e. } y^3 e^{3 x y} - x + c = 0.
\][/tex]
### Step 7: Validate the correct solution.
From these forms, we equate our derived structure:
[tex]\[
\text{Term with y-related factors} - \text{Term with x-related constants} + \text{Integration constant} = 0.
\][/tex]
Reviewing structures and integrating with hinted solutions simplifies:
[tex]\[
y^3 e^{3 x y} - x + c = 0.
\][/tex]
Thus, the solution is:
[tex]\[
\boxed{\text{e. } y^3 e^{3 x y} - x + c = 0.}
\][/tex]
