In this problem you will use variation of parameters to solve the nonhomogeneous equation

$t^2{y}^{\text{'}\text{'}}+2t{y}^{\text{'}}-6y=3t^3+3t^2$

A$.$ Plug $y=t^n$ into the associated homogeneous equation to get an equation with only $t$ and $n.$

B$.$ Solve the equation above for $n($use $t0$ to cancel out the $t).$

You should get two values for $n,$ which give two fundamental solutions of the form $y=t^n.$

$y_1=$

$W(y_1,y_2)=$

C$.$ To use variation of parameters, the linear differential equation must be written in standard form

$y^{\text{'}\text{'}}+p{y}^{\text{'}}+qy=f(t)$

What is the function $f(t)?$

$f(t)=$

D$.$ Compute the following integrals.

$u_1={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{-y_{2}f(t)}{W}dt=$

$u_2={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{y_{1}f(t)}{W}dt=$

E$.$ Write the general solution. $($Use $c1$ and $c2$ for $c_1$ and $c_2).$

$y=$