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Were you able to find the complimentary solution, \(\large y_c\) ?

yes

\(y = y_h + y_c\)

\[y_p = ke^{-x}cosx\]
\[dy/dp = -ke^{-x}cosx -ke^{-x}sinx\]
\[d^2y/dp^2 = 2ke^{-x}sinx\]

dy/dx and d^2y/dx^2 obviously lol

k = 1/3; my bad

check if the the auxillary equation is
2D^2 +4D+7=0

Because the Auxiliary Equation gave me these roots,\[\large \lambda =-1\pm\frac{\sqrt{10}}{2}i\]