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Math Help - Proving the general solution of a differential equation

  1. #1
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    Proving the general solution of a differential equation

    Prove that the general solution to the differential equation, \frac{dy}{dx}=\frac{k}{xhy} is y^2=lnCx^n where k, h, C and n are constants.

    \frac{dy}{dx}=\frac{k}{xhy}

    \int{y}dy=\int\frac{A}{x}dx where A=\frac{k}{h}

    \frac{1}{2}y^2=Alnx+C

    how can I twit the equation further?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Proving the general solution of a differential equation

    Quote Originally Posted by Punch View Post
    \frac{1}{2}y^2=Alnx+C how can I twit the equation further?
    y^2=2A\ln x+2C=\ln x^{2A}+\ln K=\ln (Kx^{2A})=\ln (Kx^n)
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    Re: Proving the general solution of a differential equation

    Quote Originally Posted by FernandoRevilla View Post
    Express the equation in the form hy\;dy-\frac{kdx}{x}=0 (separated variables) .
    Alright, then i get h\frac{y^2}{2}=klnx + C which I still dont see how I can further transform this
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    Re: Proving the general solution of a differential equation

    Quote Originally Posted by FernandoRevilla View Post
    y^2=2A\ln x+2C=\ln x^{2A}+\ln K=\ln (Kx^{2A})=\ln (Kx^n)
    Oh! I see, so I just have to insert a lnC' to replace C and that does the trick!
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