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Math Help - Diffrential equations!

  1. #1
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    Diffrential equations!

    Find differential equation that satisfy all values of the arbitrary constants :
    y = Ax^2 + e^x
    The answer is
    x\frac{dy}{dx} = 2y + (x  - 2)e^x

    Right, so i start off by eliminating e^x
    Equation one : y = Ax^2 + e^x
     Equation two : \frac{dy}{dx} = 2Ax + e^x
    Equation three: \frac{d^2y}{dx^2} = 2A + e^x

    Eliminate e^x
    Minus both equations 1 & 2 and i get
    y - \frac{dy}{dx} = A(x^2 - 2x)
    Minus both equations 2 & 3 and i get
    \frac{d^2y}{dx^2} - \frac{dy}{dx} = A(2 - 2x)

    Now eliminate A
     \frac{y - \frac{dy}{dx}}{(x^2 - 2x)} - \frac{\frac{d^2y}{dx^2} - \frac{dy}{dx}}{(2 - 2x)} = 0

    SO then i cross multiply and get the wrong answer...am i doing something wrong above?

    Thank-you
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  2. #2
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    Quote Originally Posted by AshleyT View Post
    Find differential equation that satisfy all values of the arbitrary constants :
    y = Ax^2 + e^x
    The answer is
    x\frac{dy}{dx} = 2y + (x - 2)e^x

    Right, so i start off by eliminating e^x
    Equation one : y = Ax^2 + e^x
     Equation two : \frac{dy}{dx} = 2Ax + e^x
    Equation three: \frac{d^2y}{dx^2} = 2A + e^x

    Eliminate e^x
    Minus both equations 1 & 2 and i get
    y - \frac{dy}{dx} = A(x^2 - 2x)
    Minus both equations 2 & 3 and i get
    \frac{d^2y}{dx^2} - \frac{dy}{dx} = A(2 - 2x)

    Now eliminate A
     \frac{y - \frac{dy}{dx}}{(x^2 - 2x)} - \frac{\frac{d^2y}{dx^2} - \frac{dy}{dx}}{(2 - 2x)} = 0

    SO then i cross multiply and get the wrong answer...am i doing something wrong above?

    Thank-you
    y = Ax^2 + e^x .... (1)

    \Rightarrow \frac{dy}{dx} = 2Ax + e^x \Rightarrow A = \left( \frac{dy}{dx} - e^x\right) \cdot \frac{1}{2x} .... (2)

    Substitute (2) into (1) to eliminate A and the solution falls out easily.
    Last edited by mr fantastic; December 9th 2008 at 01:32 PM.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    y = Ax^2 + e^x .... (1)

    \Rightarrow \frac{dy}{dx = 2Ax + e^x \Rightarrow A = \left( \frac{dy}{dx} - e^x\right) \cdot \frac{1}{2x} .... (2)

    Substitute (2) into (1) to eliminate A and the solution falls out easily.
    Uh, there's an error in your post so i cannot read it?
    Thank-you very much for replying though .

    Is the second equation in your post meant to be dy/dx ?
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  4. #4
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    Quote Originally Posted by AshleyT View Post
    Uh, there's an error in your post so i cannot read it?
    Thank-you very much for replying though .

    Is the second equation in your post meant to be dy/dx ?
    Sorry about that. I was posting in a hurry. Fixed now.
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    Thank-you very much!
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