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Q20E

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**

**${{x}}^{{2}}{{e}}^{{x}}{-}{x}{s}{i}{n}{4}{x}{+}{{x}}^{{3}}$**

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

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

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

Then

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

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

Then

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

Let $i\left(x\right)={x}^{3}$

Then

${D}^{4}\left[i\right]=0$

Hence

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

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

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