Find a third order linear differential equation whose solutions include e^x+xe^x.
The easiest way I can think of would be as follows...
If $\displaystyle e^x + xe^x$ is a solution then
$\displaystyle y = e^x + xe^x$.
Take the derivative, you get
$\displaystyle \frac{dy}{dx} = e^x + e^x + xe^x$
$\displaystyle = e^x + y$.
Take the second derivative, you get
$\displaystyle \frac{d^2y}{dx^2} = e^x + \frac{dy}{dx}$.
Take the third derivative, you get
$\displaystyle \frac{d^3y}{dx^2} = e^x + \frac{d^2y}{dx^2}$.
So a third order linear DE that has $\displaystyle e^x + xe^x$ as a solution would be
$\displaystyle \frac{d^3y}{dx^3} - \frac{d^2y}{dx^2} = e^x$.