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Math Help - Find a differential equation given solution

  1. #1
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    Find a differential equation given solution

    Find a diŽerential equation whose solution is the two parameter fam-
    ily of curves ln
    y = c1*x^2+ c2.

    I've gotten this far:
    y= e^(c1*x^2+ c2)

    y'= 2*c1*x* (e^(c1*x^2+ c2).

    I also found y''. But for the world of it I cannot figure out the differential equation from y, y' and y''.

    Really appreciate your help!



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  2. #2
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    \ln y=C_1 x^2+C_2\qquad \bigg/ \frac{d}{dx}
    \frac{1}{y}y'=2C_1 x\qquad \bigg/ \frac{d}{dx}
    \frac{yy''-y'^{{}^2}}{y^2}=2C_1   \bigg/ \cdot y^2
    yy''-y'^{{}^2}=2C_1y^2 or if you prefer yy''-y'^{{}^2}-Cy^2=0.
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