Q8E - 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. $\mathbf{ty}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{sin}\mathbf{3}\mathbf{t}$

Step-by-Step Solution

Answer

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

Step 1: Use the method of undetermined coefficients to find a particular solution of a given differential equation.

Given equation is $\mathrm{ty}\text{'}\text{'}-\mathrm{y}\text{'}+2\mathrm{y}=\sin 3\mathrm{t}.$

 

Here, the given differential equation is non-homogeneous.

 

According to the method of undetermined coefficients, the method of undetermined coefficients applies to the differential equation of the following form,

 

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

 

Here, a, b and c are the constants.

Step 2: Final conclusion.

The coefficient of $y\text{'}\text{'}$ is $t$ which is not constant. Therefore, we can say that L.H.S. of the given equation is not a differential equation with a constant coefficient.

 

Thus, the method of undetermined coefficients cannot be applied.