1. The problem asks to identify the correct way to rewrite the differential equation $$3x \frac{dx}{dt} - x^{2} + t^{2} = \ln(t)$$ for solving by separation of variables. 2. Separation of variables requires rewriting the equation so that all terms involving $x$ are on one side and all terms involving $t$ are on the other side, typically in the form $$f(x) dx = g(t) dt$$. 3. Starting from the given equation: $$3x \frac{dx}{dt} - x^{2} + t^{2} = \ln(t)$$ Rearrange to isolate $$\frac{dx}{dt}$$: $$3x \frac{dx}{dt} = x^{2} - t^{2} + \ln(t)$$ $$\frac{dx}{dt} = \frac{x^{2} - t^{2} + \ln(t)}{3x}$$ 4. To separate variables, we want to write it as: $$\frac{dx}{dt} = h(x) + k(t)$$ But here, the right side mixes $x$ and $t$ terms in a way that cannot be separated into a product of a function of $x$ and a function of $t$. 5. Checking the options: - Option A: $$\int 3x dx - \int x^{2} dx = \int (\ln(t) - t^{2}) dt$$ - Option B: $$\int (3x - x^{2}) dx = \int (\ln(t) - t^{2}) dt$$ - Option C: $$\int \frac{3x}{x^{2}} dx = \int (\ln(t) - t^{2}) dt$$ - Option D: The differential equation cannot be solved by separation of variables. 6. Since the equation cannot be rearranged into a product of a function of $x$ and a function of $t$, separation of variables is not applicable. Final answer: Option D.