Find ODE from given solutions I know this is an easy question. However, I need a little bit of help to start.
If $e^{3x}$ and $e^{-x}$ are solutions of a second order constant coefficient linear homogeneous ode, what is the ode?

We want to look for solutions of the form $y=ce^{\lambda x}$. 
Subbing this into the general ODE of the form $y'' + ay'+by=0$ gives
$$ (\lambda^2 + a\lambda +b)ce^\lambda = 0$$
But of course $ce^\lambda \neq 0$, so we have
$$\lambda^2 + a\lambda +b = 0\tag{1}$$
where $\lambda = -1,3$. 
Subbing in $\lambda = 3,-1$ into $(1)$ gives 
$$ 3a+b=-9$$
and
$$ a-b=1$$
Solving gives $$ a=-2,  b=-3$$
