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Thread: Differential equation and homogeneous equation of degree n

  1. #1
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    Differential equation and homogeneous equation of degree n

    A function f of two variables is said to be homogeneous of degree n if

    f(tx,ty)=tnf(x,y)

    whenever t>0.

    Show that such a function f satisfies the equation

    x(df/dx)=y(df/dy)=nf.

    **('d's in above equation supposed to be curly d, but I couldn't figure out how to insert them)**
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  2. #2
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    Re: Differential equation and homogeneous equation of degree n

    What you have stated here is not true! f(x, y)= x^2+ y^2 is homogeneous of order 2: f(tx, ty)= (tx)^2+ (ty)^2= t^2x^2+ t^2y^2= t^2(x^2+ y^2)= t^2f(x, y). But x\frac{\partial f}{\partial x}= 2x^2 and y\frac{\partial f}{\partial y}= 2y^2. They are not equal and not equal to 2f.

    Did you mean x\frac{\partial f}{\partial x}+ y\frac{\partial f}{\partial y}= n f?
    Last edited by HallsofIvy; Aug 13th 2017 at 03:49 AM.
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  3. #3
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    Re: Differential equation and homogeneous equation of degree n

    Sorry, that's exactly what I meant - the '=' was a typo.
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