Q11E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{4}\mathbf{sin}\mathbf{3}\mathbf{t}\mathbf{-}{\mathbf{t}}^2{\mathbf{e}}^{3t}\mathbf{+}\frac{1}{t}$

Step-by-Step Solution

Answer

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

Step 1: Use the method of undetermined coefficients

Given equation,

 

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{4}\mathbf{sin}\mathbf{3}\mathbf{t}\mathbf{-}{\mathbf{t}}^2{\mathbf{e}}^{3t}\mathbf{+}\frac{1}{t}$

 

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, sine, cosine, or products of these functions. 

Step 2: Conclusion

The given R.H.S. of the equation $\frac{1}{\mathrm{t}}$ is not the combination of polynomials t. 

 

From the definition of a polynomial, it should have only non-negative integers as powers. 

 

Therefore, it is not a product of a polynomial and an exponential function. 

 

So, the method of undetermined coefficients cannot be applied.