Differential equations.

**AmyZheng** · December 17th 2008, 12:00 AM

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

**Prove It** · December 17th 2008, 12:32 AM

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

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

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