设f(x+y,xy)=x^2y+y^3,则f(x,y)的表达式为

Question

设f(x+y,x_y)=x^2y+y^3,则f(x,y)的表达式为

Answer 1

由于x=[(x+y)+(x-y)]/2,y=[(x+y)-(x-y)]/2
所以f(x+y,x-y)=x²y+y³=y(x²+y²)={[(x+y)-(x-y)]/2}.{{[(x+y)+(x-y)]/2}²+{[(x+y)-(x-y)]/2}²}
所以f(x,y)=(x-y)/2.{[(x+y)/2]²+[(x-y)/2]²}=(x-y)/2.(x²+y²)/2=(x-y)(x²+y²)/4