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Q5E

Expert-verifiedFound in: Page 180

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.

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.

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**.**

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