Çözüldü Differential Equation - Quadratic Equation with Complex Roots

A question with no solution or answer from Stanford University:

Find a second-order linear, homogeneous equation with constant real coefficients that has y(t) = e^(2t)·sin(t) as a solution. What is the general solution of this equation?
http://math.stanford.edu/~ralph/math53/Site/Math53_files/53mid1_s06.pdf (The last problem)

Based on the solution y(t) = e^(2·t)·sin(1·t), the complex roots of the auxiliary (characteristic) equation must be r = 2 ∓ 1·i, so this equation is;
[ r - (2 + i) ][ r - (2 - i) ] = 0
[ (r - 2) - i ][ (r - 2) + i ] = 0
(r - 2)^2 - i^2 =
r^2 - 4r + 4 - (-1) = 0
r^2 - 4r + 5 = 0 and converting back to the differential equation y'' - 4y' + 5 = 0
A, B ∈ R
y = e^(2t)·(A·cos(t) + B·sin(t)).

Check by WolframAlpha: https://www.wolframalpha.com/input?i=y'' - 4y' + 5y = 0
(Note: x = t)

Another harder solution based on the fact that "different differential equations can have the exact same particular solution."

https://i72.servimg.com/u/f72/19/97/10/39/de_gem10.png

y = e^(2t)·sin(t)
y' = 2e^(2t)·sin(t) + e^(2t)·cos(t)
y' - 2y = e^(2t)·cos(t)
y'' - 2y' = 2e^(2t)·cos(t) - e^(2t)·sin(t)
y'' - 2y' + y = 2e^(2t)·cos(t)

Solution by Google Gemini AI:
(Lagrange Variation of Coefficients Method can also be applied, and it is homework for interested students.)

https://i72.servimg.com/u/f72/19/97/10/39/de_gem11.png