Q12E - 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. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{=}{\mathbf{e}}^t\mathbf{+}\mathbf{7}$

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,

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{=}{\mathbf{e}}^t\mathbf{+}\mathbf{7}$

 

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. 

 

And the left-hand side consists of the differential equation with constants coefficients.

Step 2: Conclusion.

The L.H.S. of the equation has a non-constant coefficient, $\mathrm{t}$. 

 

Thus, the method of undetermined coefficients cannot be applied.