设F(xy,y^2/x)=x^2+y^2,求F(y^2/x,xy)

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设F(xy,y^2/x)=x^2+y^2,求F(y^2/x,xy)
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设F(xy,y^2/x)=x^2+y^2,求F(y^2/x,xy)
设F(xy,y^2/x)=x^2+y^2,求F(y^2/x,xy)

设F(xy,y^2/x)=x^2+y^2,求F(y^2/x,xy)
F(xy,y^2/x)=x^2+y^2,
令m=xy,n=y^2/x,则:
mn=y^3,
y^2=(mn)^(2/3);
m/n=x^2/y,
x^2=m/n*y=m/n*(mn)^(1/3)=m^(4/3)*n^(-2/3).
所以F(m,n)=(mn)^(2/3)+m^(4/3)*n^(-2/3),
所以 F(n,m)=(nm)^(2/3)+n^(4/3)*m^(-2/3),
F(y^2/x,xy)=y^2+(y^2/x)^(4/3)*(xy)^(-2/3)=y^2+y^2/x^2.