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Q16E

Expert-verifiedFound in: Page 337

Book edition
9th

Author(s)
R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages
616 pages

ISBN
9780321977069

**find a differential operator that annihilates the given function.**

${{x}}^{{2}}{-}{{e}}^{{x}}$

${D}^{4}-{D}^{3}$is the differential operator that annihilates the given function.

Let the function be $f\left(x\right)={x}^{2}-{e}^{x}$

Let $g\left(x\right)={x}^{2}$

Then

$g\text{'}\left(x\right)=2x\phantom{\rule{0ex}{0ex}}g\text{'}\text{'}\left(x\right)=2\phantom{\rule{0ex}{0ex}}g\text{'}\text{'}\text{'}\left(x\right)=0\phantom{\rule{0ex}{0ex}}{D}^{3}\left[g\right]=0$

Let $h\left(x\right)=-{e}^{x}$

Then

$h\text{'}\left(x\right)=-{e}^{x}\phantom{\rule{0ex}{0ex}}h\text{'}\left(x\right)-h\left(x\right)=0\phantom{\rule{0ex}{0ex}}(D-1)\left[h\right]=0$

Hence ${D}^{3}(D-1)\left[f\right]=0$

Then ${D}^{4}-{D}^{3}$is the differential operator that annihilates the given function.

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