Which of the following functions xt 3et et e2t xt 3et et

Which of the following functions x(t) = (3e^t + e^-t, e^2t) x(t) = (3e^t + e^-t, e^t) x(t) = (3e^t + e^-t, te^t) x(t) = (3e^t, t^2 e^t) x(t) = (e^t + 2e^-t, e^t + 2e^-t) can be solutions of a first-order autonomous homogeneous system? (Compare with the necessary structure of the solution found in this section.)

Solution

If we take second order homo diff

Suppose its auxilary equation is D^2-2D+4=0

(D-2)^2=0

D=2,2 Solut is ( c1+c2x)e^2x

In case if D=2,-2 then c1e^2x+c2e^-2x

If order is one then D has only one value

Henec we can conclude that solution of the diff equation except order one will be the linear combination of two or more than two solution.

And in given problem all are linear combination if two solutions hence none of them are solution of first order diff equation