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Found in: Page 337

### Fundamentals Of Differential Equations And Boundary Value Problems

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

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

See the step by step solution

## Step 1: Solve the homogeneous of the given equation

The homogeneous of the given equation is

$\left({D}^{2}+2D+1\right)\left[y\right]=0$

The solution of the homogeneous is

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

Now ${x}^{2}-x+1$ is annihilated by ${D}^{3}$

Then, every solution to the given nonhomogeneous equation also satisfies

. ${D}^{3}\left({D}^{2}+2D+1\right)\left[y\right]=0$

Then

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

is the general solution to this homogeneous equation

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

Comparing (1) & (2)

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

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