f(x+y,y/x)=x²-y².求f(x,y)

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f(x+y,y/x)=x²-y².求f(x,y)

f(x+y,y/x)=x²-y².求f(x,y)
f(x+y,y/x)=x²-y².求f(x,y)

f(x+y,y/x)=x²-y².求f(x,y)
令x+y=u
y/x=v
解得x=u/(v+1)
y=uv/(v+1)
f(u,v)=f(x+y,y/x)=x²-y²=(x+y)(x-y)=u[u/(v+1)-uv/(v+1)]=-u²(1-v)/(v+1)
f(u,v)=-u²(1-v)/(v+1)
所以f(x,y)=x²(y-1)/(y+1)

令x+y=u,y/x=v;通过上面两个式子解出用u、v表示的X、y;
解之可得:x=u/(1+v),y=uv/(1+v);
即原函数为f(u,v)=x²+y²=[u/(1+v)]²+[uv/(1+v)]²
即求出f(u,v)=x²+y²=[u/(1+v)]²+[uv/(1+v)]²
将u,v换为x,y即可得到