I have a long time to do differential equations and now I am stuck with this... If anyone could post the solution I would be thankful.
Let (E) be a nth order linear homogeneous differential equation with constant coefficients. Let L(i), i=1,2,...,s be the roots of the characteristic polynomial with multiplicity N(i), i=1,2,...,s respectively. Then the following are true:
(I) All the solutions of (E) as well as their (n-1)th order derivatives are bounded in [0,+oo) if and only if for every i=1,2,...,s we have Re(L(i))<= 0 and Re(L(i))=0 => N(i)=0
(II)The limit of all solutions of (E) as well as their derivatives up to (n-1)th order is 0 as x->+oo if and only if Re(L(i))<0 for i=1,2,...,s