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    Hi, I've been stuck on this question:

    Determine for which values of the parameters a and b the following function is homogeneous:

    z(x,y) = [(x^a)(y^2b)+x(y^a)] / [x^b-2(y^a)]

    Sorry for lack of maths font.

    I've used the definition z(tx,ty)=t^k z(x,y) then I tried substituting y=x and then t=x but I still can't solve for a and b. Any suggestions please?
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    Writing out your definition of a homogeneous function using this function, we get:
    \dfrac{t^{a}x^{a}t^{2b}y^{2b} + txt^ay^a}{t^bx^b-2t^ay^a} = t^k\left[ \dfrac{x^ay^{2b} + xy^a}{x^b-2y^a} \right]

    Clearly if this is to be the case, then we need the powers of both multiples of t on the top to be equal, and the powers of t on the bottom to be equal (simplifying it first makes this easier). This gives you simultaneous equations in a,b to solve.
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    Thanks, so a+2b=a+1 for the top and a=b for the bottom?
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    (Original post by takeonme79)
    Thanks, so a+2b=a+1 for the top and a=b for the bottom?
    Yup Depending on what level you're working at, your solution might need to show that this is a necessary and sufficient condition for the function to be homogeneous.
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    Thanks very much!
 
 
 
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