Chapter 6: Q15E (page 337)

find a differential operator that annihilates the given function.
$e^{2 x} - 6 e^{x}$

Short Answer

Expert verified

$D^{2} - 3 D + 2$ is the differential operator that annihilates the given function.

Step by step solution

Differentiate the function continuously

$h (x) = - 6 e^{x}$ Let the function be $f (x) = e^{2 x} - 6 e^{x}$

Let $g (x) = e^{2 x}$

Then

$g' (x) = 2 e^{2 x} = 2 g (x) \Rightarrow g' (x) - 2 g (x) = 0 \Rightarrow (D - 2) [g] = 0$

Let $h (x) = - 6 e^{x}$

Then

$h' (x) = - 6 e^{x} = h (x) \Rightarrow h' (x) - h (x) = 0 \Rightarrow (D - 1) [h] = 0$

Then

$(D - 2) [f] (x) = (D - 2) [g + h] (x) = (D - 2) [g] (x) + (D - 2) [h] (x) = (D - 2) [h] (x) = 6 e^{x} = - h (x)$

Then

$(D - 1) [(D - 2) [f]] (x) = (D - 1) [- h] (x) = - (D - 1) [h] (x) = 0$

Hence $(D - 1) (D - 2) [f] = 0$

Then $D^{2} - 3 D + 2$ is the differential operator that annihilates the given function.

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find a differential operator that annihilates the given function.
$e^{2 x} - 6 e^{x}$

Short Answer

Step by step solution

Differentiate the function continuously

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find a differential operator that annihilates the given function.e2x-6ex

Short Answer

Step by step solution

Differentiate the function continuously

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Study anywhere. Anytime. Across all devices.

find a differential operator that annihilates the given function.
$e^{2 x} - 6 e^{x}$