函数f(xy,y^2/x)=x^2+y^2,求f(y^2/x,xy)

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函数f(xy,y^2/x)=x^2+y^2,求f(y^2/x,xy)
函数f(xy,y^2/x)=x^2+y^2,求f(y^2/x,xy)

函数f(xy,y^2/x)=x^2+y^2,求f(y^2/x,xy)
f(xy,y^2/x)=x^2+y^2
设xy=a
y^2/x=b
得
x=a^(2/3)b^(-1/3)
y=a^(1/3)b^(1/3)
得
f(a,b)=a^(2/3)b^(2/3)+a^(4/3)b^(-2/3)
所以
f(y^2/x,xy)=y^2+y^2/x^2

令xy=a,y^2/x=b,则a*b=y^3则y=(ab)^(1/3),a^2/b=x^3,x=(a^2/b)^(1/3),也就是说
f(a,b)=(a^2/b)^(2/3)+(ab)^(2/3);所以f(y^2/x,xy)=((y^2/x)^2/(xy))^(2/3)+(y^3)^(2/3)=y/x+y^2.本题的考点是掌握函数与使用字母没有关系。不明白可追问。

设u=xy,v=y^2/x
y=u/x
y^2=u^2/x^2代入v得：v=u^2/x^3
x=³√(u^2/v)
y=³√(uv)
f(u,v)=³√(u^4/v^2)+³√(u^2v^2)
f(y^2/x,xy)=f(v,u)
=³√(v^4/u^2)+³√(v^2u^2)

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设u=xy,v=y^2/x
y=u/x
y^2=u^2/x^2代入v得：v=u^2/x^3
x=³√(u^2/v)
y=³√(uv)
f(u,v)=³√(u^4/v^2)+³√(u^2v^2)
f(y^2/x,xy)=f(v,u)
=³√(v^4/u^2)+³√(v^2u^2)
=³√[(y^2/x)^4/(xy)^2]+³√[(y^2/x)^2(xy)^2]
=³√[(y^6/x^2)+³√(y^4)
=(y^2* ³√x)/x+y* ³√y
思路大概如果，结果请自己演算

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