Solving 2nd Order Linear DE with Constant Coefficients

[Apteronotus]

Hi,

When solving a 2nd order Linear DE with constant coefficients (ay''+by'+cy=0) we are told to look for solutions of the form y=e^{rt} and then the solution (if we have 2 distinct roots of the characteristic) is given by
y(t)=c_1 e^{r_1 t}+c_2 e^{r_2 t}

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?

 

[tiny-tim]

Hi Apteronotus!

… how do we know there are no other solutions?

It's easy to prove for the first-order case …

if y' - ry = 0, put y = ze^rt, then (z' + rz)e^rt = rze^rt

so e^rt = 0 (which is impossible),

or z' + rz = rz, ie z' = 0, ie z is constant

and now try (y' - ry)(y' - sy) = 0, using the same trick twice ​