TITLE: For what functions is $y'' = y$?
QUESTION [8 upvotes]: What functions $y = f(x)$ have the property that $f(x) = f''(x)$, i.e. what functions have the same integral and derivitive?
I could think of $ce^x$ and $ce^{-x}$ (where $c$ is a constant), but are there others? If not, how can you prove those are the only ones?

REPLY [2 votes]: Using the differential operator $D$, the equation is
$$D^2y=y\text{, or }(D^2-1)y=0.$$
This factors as 
$$(D-1)(D+1)y=0.$$
We first set $z=(D+1)y$ and solve
$$(D-1)z=0.$$
$$Dz-z=0\implies (Dz)e^{-x}-ze^{-x}=(Dz)e^{-x}+zD(e^{-x})=D(ze^{-x})=0\implies ze^{-x}=C_0,$$
$$z=C_0e^x.$$
Then solve
$$(D+1)y=C_0e^x.$$
$$Dy+y=C_0e^x\implies (Dy)e^x+ye^x=(Dy)e^x+yD(e^x)=D(ye^x)=C_0e^{2x}\\\implies ye^x=C_1e^{2x}+C_2,$$
$$\color{green}{y=Ce^x+C'e^{-x}}.$$
The method generalizes to arbitrary polynomials in $D$.