Q4E - 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. $\mathbf{x}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{x}\mathbf{\text{'}}\mathbf{-}\mathbf{3}\mathbf{x}\mathbf{=}{\mathbf{3}}^t$

Step-by-Step Solution

Answer

Yes, the method of undetermined coefficients can be applied.

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,

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

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;

 

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

 

Compare with the given differential equation,

 

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

 

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.