Finding solution basis of $y^{(4)}-2y'''+5y''-8y'+4y=0$ Find a real-valued solution basis of $$y^{(4)}-2y'''+5y''-8y'+4y=0.$$
The corresponding characteristic equation is $$x^4-2x^3+5x^2-8x+4=0$$ $$\iff(x-1)^2(x^2+4)=0$$ which has the zeros $1, 2i, -2i$. How do I proceed from here? Please share a hint with me.
Thank you.

Hint: the multiplicity of the root $x=1$ is $2$; this means that the functions
$$y_i(x)=x^i\exp(1\cdot x)$$
with $i=0,1$ are solutions of the ODE in the OP.
