Question 9 Find a linear differential operator that annihilates the function \( x^{3}(1+7 x) \) \( O D^{2}+1 \) \( O D^{3} \) \( O D^{4} \) \( O D^{3}+2 \) \( O D^{5} \)

To determine the appropriate linear differential operator that annihilates the function \( x^{3}(1+7 x) \), we first need to find the highest derivative that produces a zero function when applied. The function \( x^{3}(1+7 x) \) is a polynomial of degree 4. Generally, a linear differential operator of order \( n \) can annihilate a polynomial of degree \( m \) if \( n \) is greater than \( m \). In this case, since \( m = 4 \), any operator of order \( 5 \) or higher will annihilate it. Hence, \( O D^{5} \) is a suitable choice. To explore this further, if you take derivatives of the original function, you would find that going beyond the fourth derivative yields zero, aligning perfectly with our findings. This polynomial remains interesting as it showcases how differential operators interact with polynomial expressions in calculus, allowing for a deeper exploration of function behavior. As for real-world applications, differential operators play a crucial role in engineering, especially in systems modeling. For example, they are extensively used in control theory to design systems that can respond predictively to various factors, ensuring stability and efficiency in processes like automated manufacturing or robotics. Understanding how linear operators function opens the door to advanced designs that make these technologies not just possible, but reliable and effective!