# 2nd order Linear DE

#### 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?

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#### tiny-tim

Homework Helper
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 = zert, then (z' + rz)ert = rzert

so ert = 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

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