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Found in: Page 180

### 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

# Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation.${\mathbf{8}}{\mathbf{z}}{\mathbf{\text{'}}}\left(\mathbf{x}\right){\mathbf{-}}{\mathbf{2}}{\mathbf{z}}\left(\mathbf{x}\right){\mathbf{=}}{\mathbf{3}}{{\mathbf{x}}}^{100}{{\mathbf{e}}}^{4x}{\mathbf{cos}}{\mathbf{25}}{\mathbf{x}}$

Yes, the method of undetermined coefficients can be applied.

See the step by step solution

## Step 1: Use the method of undetermined coefficients.

Given equation,

$8\mathrm{z}\text{'}\left(\mathrm{x}\right)-2\mathrm{z}\left(\mathrm{x}\right)=3{\mathrm{x}}^{100}{\mathrm{e}}^{4\mathrm{x}}\mathrm{cos}25\mathrm{x}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}.....\left(1\right)$

According to the method of undetermined coefficients,

To find a particular solution to the differential equation;

$\mathrm{ay}\text{'}\text{'}+\mathrm{by}\text{'}+\mathrm{cy}=\left\{\begin{array}{l}{\mathrm{Ct}}^{\mathrm{m}}{\mathrm{e}}^{\mathrm{\alpha t}}\mathrm{cos\beta t}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{or}\\ {\mathrm{Ct}}^{\mathrm{m}}{\mathrm{e}}^{\mathrm{\alpha t}}\mathrm{sin\beta t}\end{array}\right\}$

For, $\mathrm{\beta }\ne 0$ use the form:

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{t}}^{\mathrm{s}}\left({\mathrm{A}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{A}}_{1}\mathrm{t}+{\mathrm{A}}_{0}\right){\mathrm{e}}^{\mathrm{\alpha t}}\mathrm{cos\beta t}+{\mathrm{t}}^{\mathrm{s}}\left({\mathrm{B}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{B}}_{1}\mathrm{t}+{\mathrm{B}}_{0}\right){\mathrm{e}}^{\mathrm{\alpha t}}\mathrm{sin\beta t}$

## Step 2: Final conclusion.

Compare with the given differential equation,

$8\mathrm{z}\text{'}\left(\mathrm{x}\right)-2\mathrm{z}\left(\mathrm{x}\right)=3{\mathrm{x}}^{100}{\mathrm{e}}^{4\mathrm{x}}\mathrm{cos}25\mathrm{x}$

We have,

$\mathrm{\beta }=25\ne 0$

Therefore, the method of undetermined coefficients can be applied.