Find a linear differential operator that annihilates the given function. (Use $D$ for the differential operator.)

$12x + 8x^2 - \sin 2x$

Neetesh Kumar | November 05, 2024

Differential Equation Homework Help

This is the solution to Math 2A, section 13Z, Fall 2023 | WebAssign
Math002ACh4Sec05 (Homework) Question - 6
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Step-by-Step-Solution:

We need to consider separate operators for each part of the function to find a differential operator $L$ that annihilates $12x + 8x^2 - \sin 2x$.

1. Annihilation of the Polynomial Term $12x + 8x^2$:

   • For any polynomial of degree $n$, the differential operator $D^{n+1}$ will annihilate it.
   • Here, the highest degree in the polynomial $12x + 8x^2$ is $2$, so the operator $D^3$ will annihilate it: $D^3(12x + 8x^2) = 0$

2. Annihilation of the Trigonometric Term $\sin 2x$:

   • For functions of the form $\sin(ax)$ or $\cos(ax)$, the operator $D^2 + a^2$ will annihilate them.
   • In this case, $a = 2$, so the operator $D^2 + 4$ will annihilate $\sin 2x$: $(D^2 + 4)\sin 2x = 0$

3. Combine the Operators:

   • To annihilate the entire function $12x + 8x^2 - \sin 2x$, we need to take the product of the operators that annihilate each term. Therefore, the operator $L$ that annihilates $12x + 8x^2 - \sin 2x$ is: $L = D^3(D^2 + 4)$

Final Answer

The linear differential operator that annihilates $12x + 8x^2 - \sin 2x$ is:

$L = \boxed{D^3(D^2 + 4)}$

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