Second order ODE's, proof of solution method
When solving second order ODE's, we always assume that all solutions will be proportional to e^rt.
How hard is it to prove that this has to be the case?
If there's a simple proof, please don't just state it. In stead give me a hint of where to begin, so I can try to do it myself.
If the proof is long and complicated, I don't need to know all the details (because I wouldn't understand them anyway). But it still would be nice to get a brief outline of what it involves.
Re: Second order ODE's, proof of solution method
Here's one way but I don't know if it's a proof or not. Consider the ODE
y' + aby = 0,\;\;\; a\ne b \;\;(1))
where 
and 
are constant. I'm assuming that you meant constant coefficients otherwise the solutions are not usually in the form 
. Now we will factor this equation using operators. We can re-write )
as
\left(\dfrac{d}{dt} - b\right) y = 0. \;\; (2))
.
If we let
 y)
then (2) becomes
u = 0, (2))
.
or
.
We solve (it's separable) giving
Now
 y = u = c_1 e^{at})
or
This is linear which we integrate giving
noting that I have absorbed some constants into 
.
