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Q30E

Expert-verifiedFound in: Page 337

Book edition
9th

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

Pages
616 pages

ISBN
9780321977069

**use the annihilator method to determinethe form of a particular solution for the given equation.**

${y}{\text{'}}{\text{'}}{\text{'}}{+}{2}{y}{\text{'}}{\text{'}}{-}{y}{\text{'}}{-}{2}{y}{=}{{e}}^{{x}}{-}{1}$

${y}_{p}\left(x\right)={c}_{1}+{c}_{5}x{e}^{x}$

The homogeneous of the given equation is

$({D}^{3}+2{D}^{2}-D-2)\left[y\right]=0$

The solution of the homogeneous is

${y}_{h}\left(x\right)={c}_{1}{e}^{-2x}+{c}_{2}{e}^{-x}+{c}_{3}{e}^{x}$ (1)

Now ${e}^{x}-1$ is annihilated by $\left({D}^{2}-D\right)$

Then, every solution to the given nonhomogeneous equation also satisfies

. $\left({D}^{2}-D\right)$ $({D}^{3}+2{D}^{2}-D-2)\left[y\right]=0$

Then

$y\left(x\right)={c}_{1}+{c}_{2}{e}^{-2x}+{c}_{3}{e}^{-x}+{c}_{4}{e}^{x}+{c}_{5}x{e}^{x}$ (2)

is the general solution to this homogeneous equation

We know $u\left(x\right)={u}_{h}+{u}_{p}$

Comparing (1) & (2)

${y}_{p}\left(x\right)={c}_{1}+{c}_{5}x{e}^{x}$

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