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. $\mathbf{3}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{8}\mathit{y}\mathbf{=}{\mathit{t}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathit{t}\mathbf{-}{\mathit{t}}^{\mathbf{2}}{\mathit{e}}^{\mathit{t}}\mathit{s}\mathit{i}\mathit{n}\mathit{t}$

Short answer

Yes, the method of undetermined coefficient together with superposition can be applied.

Step-by-step solution

Use the method of undetermined coefficients

Given equation,

$3y\text{'}\text{'}+2y\text{'}+8y=t^2+4t-t^2{e}^{t}sint$

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, sines or cosines, or products of these functions.

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

Final conclusion.

The given,

R.H.S. of the equation $t^2$, $4t$and$t^2{e}^{t}sint$ is the combination of polynomials, exponentials, sines or cosines or product of these t function.

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

Therefore, it is a product of a polynomial and an exponential function.

So, the method of undetermined coefficients can be applied.