The characteristic equation is \(\displaystyle r^{4}-4r^{3}+8r^{2}-8r+4 = (r^{2}-2r+2)^{2} = 0 \)

which has roots \(\displaystyle 1 \pm i \), both of multiplicity 2

so the general solution is \(\displaystyle y(x) = C_{1}e^{x} \cos(x) + C_{2} e^{x} \sin x + C_{3} e^{x}x \cos x + C_{4} e^{x}x \sin x \)