(Solved):     A gth order, linear, homogeneous, constant coefficient differential equation has ...

 

 

A gth order, linear, homogeneous, constant coefficient differential equation has a characteristic equation which factors as follows. \[ \left(r^{2}-4 r+8\right)^{2} r^{2}(r+3)^{3}=0 \] Write the nine fundamental solutions to the differential equation as functions of the variable \( t \). \[ \begin{array}{lll} y_{1}= & y_{2}= & y_{3}= \\ y_{4}= & y_{5}= & y_{6}= \\ y_{7}= & y_{8}= & y_{9}= \end{array} \] (You can enter your answers in any order.) In this problem you will use variation of parameters to solve the nonhomogeneous equation \[ y^{\prime \prime}+6 y^{\prime}+9 y=-4 e^{-3 t} \] A. Write the characteristic equation for the associated homogeneous equation. (Use \( r \) for your variable.) B. Write the fundamental solutions for the associated homogeneous equation and their Wronskian. \[ \begin{aligned} y_{1} &=\\ \mathrm{W}\left(y_{1}, y_{2}\right) &= \end{aligned} \] C. Compute the following integrals. \[ \begin{array}{l} \int \frac{y_{1} g}{W} d t= \\ \int \frac{y_{2} g}{W} d t= \end{array} \] D. Write the general solution. (Use \( c 1 \) and \( c 2 \) for \( c_{1} \) and \( c_{2} \) ). \[ y= \] (Note: Your general solution will only be correct if it is a general solution to the differential equation.)