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When solving a 2nd order Linear DE with constant coefficients ([itex]ay''+by'+cy=0[/itex]) we are told to look for solutions of the form [itex]y=e^{rt}[/itex] and then the solution (if we have 2 distinct roots of the characteristic) is given by

[itex]y(t)=c_1 e^{r_1 t}+c_2 e^{r_2 t}[/itex]

This is clearly a solution, but how do we know there are no other solutions?

That is, how do we know this is the general solution?