If x\left(a-b+\frac{ab}{a-b}\right)=y\left(a+b-\frac{ab}{a+b}\right) and x+y=2a^{3} then what is x-y equal to?

Explanation

Simplifying the given terms yields x\left(\frac{a^2-ab+b^2}{a-b}\right) = y\left(\frac{a^2+ab+b^2}{a+b}\right). Cross-multiplying by a^2-b^2 gives x(a^3+b^3) = y(a^3-b^3). Let x = k(a^3-b^3) and y = k(a^3+b^3). Given x+y = k(2a^3) = 2a^3, we get k=1. Hence, x-y = (a^3-b^3) - (a^3+b^3) = -2b^3.

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