Decide whether or not the method of undetermined coefficients can be applied to find a particular solution to the given equation.$\mathbf{5}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}{\mathbf{t}}^3\mathbf{cos}\mathbf{4}\mathbf{t}$

Given equation,

$5\mathrm{y}\text{'}\text{'}-3\mathrm{y}\text{'}+2\mathrm{y}={\mathrm{t}}^3\cos 4\mathrm{t}\dots \left(1\right)$

Write the homogeneous differential equation of the equation (1),

$5\mathrm{y}\text{'}\text{'}-3\mathrm{y}\text{'}+2\mathrm{y}=0$

The auxiliary equation for the above equation,

$5{\mathrm{m}}^2-3\mathrm{m}+2=0$

Solve the auxiliary equation,

$\begin{array}{c}5{\mathrm{m}}^2-3\mathrm{m}+2=0\\ \mathrm{m}=\frac{-(-3)\pm \sqrt{9-4\left(5\right)\left(2\right)}}{2\left(5\right)}\\ \mathrm{m}=\frac{3\pm \sqrt{-31}}{10}\\ \mathrm{m}=\frac{3\pm \mathrm{i}\sqrt{31}}{10}\end{array}$

The roots of the auxiliary equation are,

${\mathrm{m}}_1=\frac{3+\mathrm{i}\sqrt{31}}{10},{\mathrm{m}}_2=\frac{3-\mathrm{i}\sqrt{31}}{10}$

The complementary solution of the given equation is,

${\mathrm{y}}_{\mathrm{c}}\left(\mathrm{x}\right)={\mathrm{e}}^{\frac{3}{10}\mathrm{x}}\left[{\mathrm{c}}_1\cosh \left(\frac{\sqrt{31}}{10}\right)+{\mathrm{c}}_2\sinh \left(\frac{\sqrt{31}}{10}\right)\right]$

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 R.H.S. of the differential equation has a finite family.

And the given R.H.S. of the equation ${\mathrm{t}}^3\cos \left(4\mathrm{t}\right)$has a final family.

The particular solution of equation (1),

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)=({\mathrm{At}}^3+{\mathrm{Bt}}^2+\mathrm{Ct}+\mathrm{D})\cos \left(4\mathrm{t}\right)+({\mathrm{Et}}^3+{\mathrm{Ft}}^2+\mathrm{Gt}+\mathrm{H})\sin \left(4\mathrm{t}\right)$

Therefore, the method of undetermined coefficients can be applied.