1. The problem is to identify the correct differential operator $L$ that represents the given differential equation: $$4 \frac{d^2 x}{dt^2} = \sin(t) \frac{dx}{dt} + 3t \cdot x = t^2$$ 2. A differential operator $L$ acts on the function $x(t)$ and includes derivatives and coefficients from the differential equation. 3. Rearranging the equation to standard form: $$4 \frac{d^2 x}{dt^2} - \sin(t) \frac{dx}{dt} - 3t \cdot x = t^2$$ 4. The operator $L$ is the left-hand side without the forcing term $t^2$: $$L = 4 \frac{d^2}{dt^2} - \sin(t) \frac{d}{dt} - 3t$$ Note that $L$ acts on $x$, so the term $-3t \cdot x$ is part of the operator. 5. Among the options, option B is: $$L \equiv 4 \frac{d^2}{dt^2} - \sin(t) \frac{d}{dt} + 3t$$ This has a plus sign before $3t$, but the equation has $-3t \cdot x$ on the left side after rearrangement. 6. Option C includes $3t \cdot x$ explicitly, which is not standard for an operator definition (operators act on functions, so $x$ is not part of the operator itself). 7. The correct operator should be: $$L \equiv 4 \frac{d^2}{dt^2} - \sin(t) \frac{d}{dt} - 3t$$ which matches the rearranged equation. 8. Since this exact option is not listed, the closest correct form is option B if we consider the sign convention, but strictly the operator includes $-3t$. Final answer: The differential operator is $$L \equiv 4 \frac{d^2}{dt^2} - \sin(t) \frac{d}{dt} - 3t$$.