Find ODE from given solutions

Question

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?

Answer 1

Hint: $3$ and $-1$ are roots of what quadratic equation?

Answer 2

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 $$ a=-2, b=-3$$