Chapter 6: Q20E (page 337)

find a differential operator that annihilates the given function
$x^{2} e^{x} - x s i n 4 x + x^{3}$

Short Answer

Expert verified

$D^{4} (D - 1)^{3} {(D^{2} + 16)}^{2}$ is the differential operator that annihilates the given function.

Step by step solution

Any nonhomogeneous term of the form f(x)=xkeαxcosβx OR role="math" localid="1663946799201" xkeαxsinβxsatisfiesrole="math" localid="1663946761280" (D-α)2+β2m[f]=0for K=0,1,2,...,m-1

Let the function be $f (x) = x^{2} e^{x} - x \sin 4 x + x^{3}$

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

Then

$(D - 1)^{3} [g] = 0$

Let $h (x) = x e^{- 5 x} \sin 3 x$

Then

${(D^{2} + 4^{2})}^{2} [h] = 0$

Let $i (x) = x^{3}$

Then

$D^{4} [i] = 0$

Hence

$(D - 1)^{3} {(D^{2} + 16)}^{2} D^{4} [f] = 0 \Rightarrow D^{4} (D - 1)^{3} {(D^{2} + 16)}^{2} [f] = 0$

Then $D^{4} (D - 1)^{3} {(D^{2} + 16)}^{2}$ is the differential operator that annihilates the given function.

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find a differential operator that annihilates the given function
$x^{2} e^{x} - x s i n 4 x + x^{3}$

Short Answer

Step by step solution

Any nonhomogeneous term of the form f(x)=xkeαxcosβx OR role="math" localid="1663946799201" xkeαxsinβxsatisfiesrole="math" localid="1663946761280" (D-α)2+β2m[f]=0for K=0,1,2,...,m-1

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find a differential operator that annihilates the given functionx2ex-xsin4x+x3

Short Answer

Step by step solution

Any nonhomogeneous term of the form f(x)=xkeαxcosβx OR role="math" localid="1663946799201" xkeαxsinβxsatisfiesrole="math" localid="1663946761280" (D-α)2+β2m[f]=0for K=0,1,2,...,m-1

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Calculus

Applied Mathematics

Pure Maths

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

find a differential operator that annihilates the given function
$x^{2} e^{x} - x s i n 4 x + x^{3}$