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Math Help - Differential equations.

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    Differential equations.

    Find a third order linear differential equation whose solutions include e^x+xe^x.
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    Quote Originally Posted by AmyZheng View Post
    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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