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

Short answer

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

Step-by-step solution

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.

Conclusion.

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

Thus, the method of undetermined coefficients cannot be applied.