Q7E - 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.$\mathbf{8}\mathbf{z}\mathbf{\text{'}}\left(\mathbf{x}\right)\mathbf{-}\mathbf{2}\mathbf{z}\left(\mathbf{x}\right)\mathbf{=}\mathbf{3}{\mathbf{x}}^{100}{\mathbf{e}}^{4x}\mathbf{cos}\mathbf{25}\mathbf{x}$

Step-by-Step Solution

Answer

Yes, the method of undetermined coefficients can be applied.

Step 1: Use the method of undetermined coefficients.

Given equation,

 

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

 

According to the method of undetermined coefficients, 

 

To find a particular solution to the differential equation;

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

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

 

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{t}}^{\mathrm{s}}({\mathrm{A}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{A}}_1\mathrm{t}+{\mathrm{A}}_0){\mathrm{e}}^{\mathrm{\alpha t}}\mathrm{cos\beta t}+{\mathrm{t}}^{\mathrm{s}}({\mathrm{B}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{B}}_1\mathrm{t}+{\mathrm{B}}_0){\mathrm{e}}^{\mathrm{\alpha t}}\mathrm{sin\beta t}$

Step 2: Final conclusion.

Compare with the given differential equation,

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

 

We have,

 

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

 

Therefore, the method of undetermined coefficients can be applied.