Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.)

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{7}\mathbf{y}\mathbf{=}{\mathbf{t}}^4{\mathbf{e}}^t$

The given differential equation is in the form of;

$\mathrm{ax}\text{'}\text{'}+\mathrm{bx}\text{'}+\mathrm{cx}={\mathrm{e}}^{\mathrm{rt}}$

According to the method of undetermined coefficients, to find a particular solution to the differential equation:

$\mathrm{ay}\text{'}\text{'}\left(\mathrm{x}\right)+\mathrm{by}\text{'}\left(\mathrm{x}\right)+\mathrm{cy}\left(\mathrm{x}\right)={\mathrm{Ct}}^{\mathrm{m}}{\mathrm{e}}^{\mathrm{rt}}$

Where m is a non-negative integer, use the form

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{t}}^{\mathrm{s}}({\mathrm{A}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{A}}_1\mathrm{t}+{\mathrm{A}}_0){\mathrm{e}}^{\mathrm{rt}}$

1. s = 0 if r is not a root of the associated auxiliary equation;
2. s = 1 if r is a simple root of the associated auxiliary equation;
3. s = 2 if r is a double root of the associated auxiliary equation.

The given differential equation is,

$\mathrm{y}\text{'}\text{'}+3\mathrm{y}\text{'}-7\mathrm{y}={\mathrm{t}}^4{\mathrm{e}}^{\mathrm{t}}......\left(1\right)$

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

$\mathrm{y}\text{'}\text{'}+3\mathrm{y}\text{'}-7\mathrm{y}=0$

The auxiliary equation for the above equation,

${\mathrm{r}}^2+3\mathrm{r}-7=0$

Solve the auxiliary equation,

$\begin{array}{c}{\mathrm{r}}^2+3\mathrm{r}-7=0\\ \mathrm{r}=\frac{-3\pm \sqrt{3^2-4\left(1\right)(-7)}}{2\left(3\right)}\\ \mathrm{r}=\frac{-3\pm \sqrt{37}}{6}\end{array}$

The roots of the auxiliary equation are;

${\mathrm{r}}_1=\frac{-3+\sqrt{37}}{6},{\mathrm{r}}_2=\frac{-3-\sqrt{37}}{6}$

The complementary solution of the given equation is;

${\mathrm{y}}_{\mathrm{c}}={\mathrm{c}}_1{\mathrm{e}}^{\frac{-3+\sqrt{37}}{6}\mathrm{t}}+{\mathrm{c}}_2{\mathrm{e}}^{\frac{-3-\sqrt{37}}{6}\mathrm{t}}$

To find a particular solution to the differential equation

$\mathrm{ay}\text{'}\text{'}\left(\mathrm{x}\right)+\mathrm{by}\text{'}\left(\mathrm{x}\right)+\mathrm{cy}\left(\mathrm{x}\right)={\mathrm{Ct}}^{\mathrm{m}}{\mathrm{e}}^{\mathrm{rt}}$

Compare with the given differential equation,

$\mathrm{y}\text{'}\text{'}+3\mathrm{y}\text{'}-7\mathrm{y}={\mathrm{t}}^4{\mathrm{e}}^{\mathrm{t}}$

Condition satisfied,

M=4, s = 0 if is not a root of the associated auxiliary equation;

Therefore, the particular solution of the equation,

$\begin{array}{c}{\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{t}}^0({\mathrm{A}}_4{\mathrm{t}}^4+{\mathrm{A}}_3{\mathrm{t}}^3+{\mathrm{A}}_2{\mathrm{t}}^2+{\mathrm{A}}_1\mathrm{t}+{\mathrm{A}}_0){\mathrm{e}}^{\mathrm{t}}\\ {\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)=({\mathrm{A}}_4{\mathrm{t}}^4+{\mathrm{A}}_3{\mathrm{t}}^3+{\mathrm{A}}_2{\mathrm{t}}^2+{\mathrm{A}}_1\mathrm{t}+{\mathrm{A}}_0){\mathrm{e}}^{\mathrm{t}}\end{array}$