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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.${\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}\left(\mathbf{\theta }\right){\mathbf{+}}{\mathbf{3}}{\mathbf{y}}{\mathbf{\text{'}}}\left(\mathbf{\theta }\right){\mathbf{-}}{\mathbf{y}}\left(\mathbf{\theta }\right){\mathbf{=}}{\mathbf{sec\theta }}$

No, the method of undetermined coefficients can’t be applied.

See the step by step solution

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

Given equation,

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{\theta }\right)\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{\theta }\right)\mathbf{-}\mathbf{y}\left(\mathbf{\theta }\right)\mathbf{=}\mathbf{sec\theta }$

Here, the given differential equation is non-homogeneous.

According to the method of undetermined coefficients, the method of undetermined coefficients applies only to non-homogeneities that are polynomials, exponentials, sines, or cosines, or products of these functions.

## Step 2: Final conclusion

The given differential equation is in the form of;

$\mathrm{ay}\text{'}\text{'}+\mathrm{by}\text{'}+\mathrm{cy}=\mathrm{g}\left(\mathrm{t}\right)$

Compare with the given differential equation,

We get,

$\mathrm{g}\left(\mathrm{t}\right)=\mathrm{sec\theta }$

We know that, $\mathrm{sec\theta }$ is not a linear combination of the term of the form of polynomials, exponentials, sine or cosine, or product of these t functions.

Therefore, the method of undetermined coefficients cannot be applied.