to the following differential equation:
$y^{(6)}-2y'''+y=0$
I obtained the following general solution:
$y=C_1e^x+C_2x e^x+C_3e^{-x/2}\cos(x\sqrt3)+C_4xe^{-x/2}\cos(x\sqrt3)+C_5e^{-x/2}\sin(x\sqrt3)+C_6xe^{-x/2}\sin(x \sqrt3)$
I would like to verify that this is correct, since when I plug the function $y$ back into the original diff eq using Mathematica I do not get zero.
The characteristic equation is $$m^6-2m^3+1=0$$ $$(m^3-1)^2=0$$ $$(m^3-1)(m^3-1)=0$$ $$(m-1)(m^2+m+1)(m-1)(m^2+m+1)=0$$ the roots are $$x_0=1$$ $$x_1=1$$ $$x_2=-\frac{1}{2}+\frac{\sqrt{3}i}{2}$$ $$x_3=-\frac{1}{2}-\frac{\sqrt{3}i}{2}$$ $$x_4=-\frac{1}{2}+\frac{\sqrt{3}i}{2}$$ $$x_5=-\frac{1}{2}-\frac{\sqrt{3}i}{2}$$
we see there are repeated roots
so the solution should be $$y=C_1e^x+C_2x e^x+C_3e^{-x/2}\cos(x\sqrt3/2)+C_4xe^{-x/2}\cos(x\sqrt3/2)+C_5e^{-x/2}\sin(x\sqrt3/2)+C_6xe^{-x/2}\sin(x \sqrt3/2)$$