$\backslash$table$[[$Entered$,$Answer Preview,Result$],[y^{\text{'}\text{'}\text{'}\text{'}}-{16}^{**}{y}^{\text{'}\text{'}\text{'}}+{70}^{**}{y}^{\text{'}\text{'}}-{136}^{**}{y}^{\text{'}}+{289}^{**}y=0,y^{\text{'}\text{'}\text{'}\text{'}}-16y^{\text{'}\text{'}\text{'}}+70y^{\text{'}\text{'}}-136y^{\text{'}}+289y=0,$incorrect$],[y=c{1}^{**}[e^{4^{**}x}{]}^{**}cos(x)+c{2}^{**}[e^{4^{**}x}{]}^{**}sin(x)+c{3}^{**}{x}^{**}[e^{4^{**}x}{]}^{**}cos(x)+c{4}^{**}{x}^{**}$

$[e^{4^{**}x}{]}^{**}sin(x),y=c_1{e}^{4x}cos(x)+c_2{e}^{4x}sin(x)+c_{3}x{e}^{4x}cos(x)+c_{4}x{e}^{4x}sin(x),$correct$]]$

At least one of the answers above is NOT correct.

Suppose that a fourth order differential equation has a solution $y=8e^{4x}xcos(x).$

$($a$)$ Find such a differential equation, assuming it is homogeneous and has constant coefficients.

$y^{\text{'}\text{'}\text{'}\text{'}}-16y^{\text{'}\text{'}\text{'}}+70y^{\text{'}\text{'}}-136y^{\text{'}}+289y=0$

help $($equations$)$

$($b$)$ Find the general solution to this differential equation. In your answer, use $c_1,c_2,c_3$ and $c_4$ to denote arbitrary constants and $x$ the independent variable. Enter $c_1$ as $c1,c_2$ as $c2,$ etc.