In this problem you will use undetermined coefficients to solve the nonhomogeneous equation $$y''+4y'+4y = 18te^{-2t}-8t$$ with initial values $\quad y(0) = 0\quad\text{and}\quad y'(0) = 4.$

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A. Write the characteristic equation for the associated homogeneous equation. (Use r for your variable.)

$\quad$

B. Write the fundamental solutions for the associated homogeneous equation.

$\quad y_1 =$

$\qquad y_2 =$

C. Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undetermined coefficients.

$\quad Y\,\,\, =$

$\quad Y'\, =$

$\quad Y'' =$

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

$\quad y =$

E. Plug in the initial values and solve for $c_1$ and $c_2$ to find the solution to the initial value problem.

$\quad y =$

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Hint: No fractions are required in the solution or answer to this problem.