1. ## Differential equations.

Find a third order linear differential equation whose solutions include e^x+xe^x.

2. Originally Posted by AmyZheng
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 $e^x + xe^x$ is a solution then

$y = e^x + xe^x$.

Take the derivative, you get

$\frac{dy}{dx} = e^x + e^x + xe^x$

$= e^x + y$.

Take the second derivative, you get

$\frac{d^2y}{dx^2} = e^x + \frac{dy}{dx}$.

Take the third derivative, you get

$\frac{d^3y}{dx^2} = e^x + \frac{d^2y}{dx^2}$.

So a third order linear DE that has $e^x + xe^x$ as a solution would be

$\frac{d^3y}{dx^3} - \frac{d^2y}{dx^2} = e^x$.