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

I am clueless. I haven't seen so many roots ever. You can say I am directly searching for full answer tbh. — Md Mosleh Uddin, Jun 20 '20 at 10:34

Moo · Answer 1 · 2020-06-20T15:21:54.050

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

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

edited Jun 20 '20 at 15:21

answered Jun 20 '20 at 12:58

Moo

11,311

Thank you sir!! – Md Mosleh Uddin Jun 20 '20 at 13:07
Done !! Thanks again – Md Mosleh Uddin Jun 20 '20 at 13:09
Please look at the ANSWER. Is the term c5 is correct? – Md Mosleh Uddin Jun 20 '20 at 13:32

Homogeneous linear DE general solution

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