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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. ${\mathbf{x}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{5}}{\mathbf{x}}{\mathbf{\text{'}}}{\mathbf{-}}{\mathbf{3}}{\mathbf{x}}{\mathbf{=}}{{\mathbf{3}}}^{t}$

Yes, the method of undetermined coefficients can be applied.

See the step by step solution

## Step 1: Use logarithms properties for simplification of the given differential equation.

Given equation,

$\mathrm{x}\text{'}\text{'}+5\mathrm{x}\text{'}-3\mathrm{x}={3}^{\mathrm{t}}$

Simplify the above equation by using logarithms properties,

$\begin{array}{c}\mathrm{x}\text{'}\text{'}+5\mathrm{x}\text{'}-3\mathrm{x}={\mathrm{e}}^{\mathrm{ln}\left({3}^{\mathrm{t}}\right)}\\ \mathrm{x}\text{'}\text{'}+5\mathrm{x}\text{'}-3\mathrm{x}={\mathrm{e}}^{\mathrm{t}\left[\mathrm{ln}\left(3\right)\right]}\\ \mathrm{x}\text{'}\text{'}+5\mathrm{x}\text{'}-3\mathrm{x}={\mathrm{e}}^{\left[\mathrm{ln}\left(3\right)\right]\mathrm{t}}\end{array}$

## Step 2: Use the method of undetermined coefficients to find a particular solution of a given differential equation.

The given differential equation is in the form of;

$\mathrm{ax}\text{'}\text{'}+\mathrm{bx}\text{'}+\mathrm{cx}={\mathrm{e}}^{\mathrm{rt}}$

According to the method of undetermined coefficients,

To find a particular solution to the differential equation;

$\mathrm{ay}\text{'}\text{'}\left(\mathrm{x}\right)+\mathrm{by}\text{'}\left(\mathrm{x}\right)+\mathrm{cy}\left(\mathrm{x}\right)={\mathrm{Ct}}^{\mathrm{m}}{\mathrm{e}}^{\mathrm{rt}}$

Where m is a non-negative integer, use the form;

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

Compare with the given differential equation,

$\mathrm{x}\text{'}\text{'}+5\mathrm{x}\text{'}-3\mathrm{x}={\mathrm{e}}^{\left[\mathrm{ln}\left(3\right)\right]\mathrm{t}}$

Condition satisfies,

s = 1 if r is a simple root of the associated auxiliary equation.

Therefore, the particular solution of the equation,

${\mathbf{y}}_{p}\left(\mathbf{x}\right)\mathbf{=}{\mathbf{Ate}}^{\mathbf{ln}\left(\mathbf{3}\right)\mathbf{t}}$

So, the method of undetermined coefficients can be applied.