Complementary Function of a 3rd Order Linear Dif. Equation of Constant Coefficients (wrong choices)

A wrong question asked in one of the exams at Izmir University of Economics, in Turkey:


https://i.servimg.com/u/f72/19/97/10/39/difden13.jpg
https://scontent.fayt2-1.fna.fbcdn....=761f26ba58c44f7b2edb1ab9794ef054&oe=60203724
https://www.facebook.com/photo?fbid=221508366247474&set=pcb.2151967974940734

All of the choices are wrong because the roots of the auxiliary equation are completely different from those given in the choices, and can be found only by Numerical Analysis, or sophisticated calculators, or referring to the computing web sites such as WolframAlpha or Symbolab:

Considering D as the derivative operator D = d / dx = r, the characteristic equation is r^3 - 2r^2 + 4 = 0 which gives one real and two complex roots as follows:
r1 ≈ -1.1304
r2 ≈ 1.5652 + 1.04343·i
r3 ≈ 1.5652 - 1.04343·i
So, the complementary function (homogenous solution) has to be;
yc(x) ≈ C1·[ e^(-1.1304·x) ] + [ e^(1.5652·x) ]·[ C1·cos(1.04343·x) + C2·sin(1.04343·x) ]

Check by WolframAlpha:

https://i72.servimg.com/u/f72/19/97/10/39/difeq_11.png
https://www.wolframalpha.com/input/...mXtWkil9Wqwe7zJSkTfC_XAY73vfgaWoLxOT-vPdK7QW8
(By clicking the "Approximate form" button)
...
Additional Comments on the correct roots and the left-hand side of the equation based on the given choices;

For A:
r1 = -2, r2 = 1 + i, r3 = 1 - i, (r + 2)(r^2 - 2r + 2) = 0,
(D^3 - 2D + 4)y = ...
Check by WolframAlpha:
https://www.wolframalpha.com/input/...oZu4MmHUqg6B-d5Nu1xACRYyEALcgh-upUMbAC8q9cH-Y

For B:
r1 = 0, r2 = 2 + i, r3 = 2 - i, r(r^2 + 4r + 5) = 0,
(D^3 + 4D^2 + 5D)y = ...
Check by WolframAlpha:
https://www.wolframalpha.com/input/...oTJw6tIS0HyRlM-OvNm9PsHhGFd4uuhU_Ou6HKEKQsw2M
Note: Coefficients are given by Wolfram arbitrarily, but the structure is correct.
(c2 - 2c1) / 5 = C1, -(c1 + 2c2) / 5 = C2

For C:
r1 = 0, r2 = -3, r3 = 5, r(r + 3)(r - 5) = 0,
(D^3 - 2D^2 - 15D)y = ...
Check by WolframAlpha:
https://www.wolframalpha.com/input/...Sekkk6P0dR6kzQvzjjmfIlhBl0kG3Z9k2R9F2GD1f6ico

For D:
Threefold repeated roots:
r1 = r2 = r3 = -2, (r + 2)^3 = 0,
(D^3 + 6D^2 + 12D + 8)y = ...
Check by WolframAlpha:
https://www.wolframalpha.com/input/...xRIv8q-7svajIBYt36ReH6tEhKLw8wafnxe6uVgy_N5OA

For E:
r1 = 0, double roots: r2 = r3 = 2, r(r - 2)^2 = 0,
(D^3 - 4D^2 + 4D)y = ...
Check by WolframAlpha:
https://www.wolframalpha.com/input/...xgSc3eBA_Ck_4J2YVDaoUgkoBIfLD7GECnKZIPIvDjO3U
(Again the same arbitrary coefficient selection, but the structure is correct.)