Q5E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{\theta }\right)\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{\theta }\right)\mathbf{-}\mathbf{y}\left(\mathbf{\theta }\right)\mathbf{=}\mathbf{sec\theta }$

Step-by-Step Solution

Answer

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

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.