Final answer:
A linear differential operator that annihilates the function e⁻¹ * 2x * eʹ - x² * eʹ is the product of the operators that annihilate each term separately, resulting in (d²/dx² - 2d/dx + 1)(d/dx - 2).
Step-by-step explanation:
The question involves finding a linear differential operator that annihilates the function e⁻¹ * 2x * eʹ - x² * eʹ. This requires us to apply differential operators to every term of the function separately.
The first term, e⁻¹ * 2x * eʹ, simplifies to 2x, which can be annihilated by the operator d/dx - 2 because its derivative is 2, which then subtracted by 2 gives zero. The second term, -x² * eʹ, can be annihilated by the operator d²/dx² - 2d/dx + 1, which is akin to applying the operator for the annihilation of x² * eʹ and it works because when you take the second derivative of x², you get 2, and then apply the remaining parts of the operator accordingly.
When combining the operators to find one that annihilates the entire function, we have to take the least common multiple of the differential operators, which is essentially the product of the two, giving us (d²/dx² - 2d/dx + 1)(d/dx - 2) as the linear differential operator that annihilates the given function.