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Find the annihilation operator for y = 7e^{-5x} + 2e^{-2x}

Question:

Find the annihilation operator for {eq}y = 7e^{-5x} + 2e^{-2x} {/eq}

Differential operators:

The differential operator {eq}D^n {/eq} cancels each of the following functions: {eq}1,x,x^2,...,x^{n-1} {/eq}

The differential operator {eq}(D-\alpha)^n {/eq} cancels each of the following functions: {eq}e^{\alpha x},xe^{\alpha x},x^2e^{\alpha x},...,x^{n-1}e^{\alpha x} {/eq}

The differential operator {eq}[D^2-2\alpha D+(\alpha^2+\beta^2)]^n {/eq} cancels each of the following functions: {eq}e^{\alpha x}\cos(\beta x),xe^{\alpha x}\cos(\beta x),x^2e^{\alpha x}\cos(\beta x),...,x^{n-1}e^{\alpha x}\cos(\beta x)\\ e^{\alpha x}\sin(\beta x),xe^{\alpha x}\sin(\beta x),x^2e^{\alpha x}\sin(\beta x),...,x^{n-1}e^{\alpha x}\sin(\beta x)\\ {/eq}

Answer and Explanation:

{eq}y = 7e^{-5x} + 2e^{-2x} {/eq}

We know that for an expression with the form {eq}Ce^{\alpha x} {/eq}, the annihilation operator is {eq}D-\alpha {/eq}

The annihilation operator for 7e^{-5x} is {eq}D+5 {/eq}


The annihilation operator for 2e^{-2x} is {eq}D+2 {/eq}


The annihilation operator for 7e^{-5x} + 2e^{-2x} is {eq}(D+5)(D+2) {/eq}


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Differential Calculus: Definition & Applications

from Calculus: Help and Review

Chapter 13 / Lesson 6
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