Second order linear homogeneous ODE with constant coefficients

In homework I was asked to find all solutions to the following ODE:$$x''+ax'+bx = 0$$After reading, I know the following.

(a)
If $t^2+at+b = (t-\lambda_1)(t-\lambda_2)$ with $\lambda_1\ne\lambda_2$, then$$x(t) = c_1e^{\lambda_1 t}+c_2e^{\lambda_2 t} $$ is the general solution.

(b)
If $t^2+at+b = (t-\lambda)^2$ then $x(t) = (c_1+c_2t)e^{\lambda t}$is the general solution.

EDIT
I know why and when these are solutions, but this is not my question. My question is how to show if $x(t)$ satisfies this equation, then it must be in one of the two forms. Not that these two forms are solutions.

EDIT
Set $y = (x,x')$, then $y' = F(y) = (y(2),-a\cdot y(2)-b\cdot y(1))$
If I know $F(y)$ is locally Lipschitz then by Picard–Lindelöf theorem solution is unique.