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Thread: Differenntial equation

  1. #1
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    Differenntial equation

    Who can tell me what's the Deffrential equation of of this general equation y=(Ae^5x)+(Bxe^5x)
    please 😊😊
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  2. #2
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    Differenntial equation

    Hi Hama

    Please note that if a differential equation has a repeated root of characteristics equation
    $\displaystyle {(r - \alpha)^2} = 0$
    then the solution will be
    $\displaystyle y = A{e^{\alpha x}} + Bx{e^{\alpha x}}$
    Thanks from romsek and HallsofIvy
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  3. #3
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    Re: Differenntial equation

    Another way (if you do not know about the characteristic equation):

    from $\displaystyle y= Ae^{5x}+ Bxe^{5x}$, $\displaystyle y'= 5Ae^{5x}+ Be^{5x}+ 5Bxe^{5x}= (5A+ B)e^{5x}+ 5Bxe^{5x}$ and $\displaystyle y''= 5(5A+ B)e^{5x}+ 5Be^{5x}+ 25Bxe^{5x}= (25A+ 10B)e^{5x}+ 25Be^{5x}$.

    Subtracting 10 times the second equation from the third, $\displaystyle y''- 10y'= (25A+ 10B)e^{5x}+ 25Be^{5x}- (50A+ 10B)e^{5x}- 50Bxe^{5x}= -(25Ae^{5x}+ 25Bxe^{5x})$. Adding 25y eliminates the arbitrary constants, A and B, and gives $\displaystyle y''- 10y'+ 25y= 0$.
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  4. #4
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    Re: Differenntial equation

    Quote Originally Posted by Hama View Post
    Who can tell me what's the Deffrential equation of of this general equation y=(Ae^5x)+(Bxe^5x)
    please ����
    There is no single differential equation for any function.
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  5. #5
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    Re: Differenntial equation

    On the other hand, here we have a family of functions, not a single function. What HaiLiang and I give is the simplest linear differential equation having every member of the given family as solution and no other solutions.
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