Homogeneous linear DE general solution

Question

I need help with a homework question. I am clueless about this. Too many roots. I don't know what to do. Please help me with the complete solution. Please ignore the definition part. Picture link

Answer 1

We are told the the roots to the auxiliary equation are

$$1, 2, 2, 2, \dfrac{1}{2} + \dfrac{\sqrt{3}}{2}, \dfrac{1}{2} - \dfrac{\sqrt{3}}{2}, 1 + i, 1 + i, 1 - i, 1 - i, 1 + 2 i, 1 - 2i$$

Since we have the roots of the auxiliary equation and their multiplicities, we can directly write the homogeneous solution

$y(t) = c_1 e^t + e^{2t}(c_2 + c_3 + c_4 t^2) +c_5 e^{\left(\frac{1}{2}+\frac{\sqrt{3}}{2}\right) t} + c_6 e^{\left(\frac{1}{2}-\frac{\sqrt{3}}{2}\right) t}+ e^t \cos t (c_7 + c_8 t) + e^t\sin t( c_9 + c_{10} t) + e^t(c_{11} \cos 2t + c_{12} \sin 2t)$