Why is the exponential function used function used when solving linear 2nd order homogeneous differential equations? In my textbook the introduction to solving linear 2nd order homogeneous DE's begins with a general form:
$ay''+by'+ cy =0$ 
Then they say: "a solution must have the property that its second derivative is expressible as a linear combination of its first and zeroth derivatives. This suggests a solution of the form:"
$y=e^{rt}$
By this doesn't seem intuitive to me. How is the second derivative of the exponential function expressible as a linear combination of its first and zeroth derivative? Would that not mean the following:
$r^2e^{rt}=re^{rt}+e^{rt}$
Which isn't true. So what could they mean by that? 

"a solution must have the property that its second derivative is expressible as a linear combination of its first and zeroth derivatives." 
"So what could they mean by that?"
If
$ay^{\prime\prime}+by^\prime+cy=0$
then
$y^{\prime\prime}=-\frac{b}{a}y^\prime-\frac{c}{a}y$
That is all that is meant by that statement.
