adding and subtracting rational expressions?

Question

x^3/ x^2-y^2 + y^3/ y^2-x^2

I'm having trouble with this concept, I'll show you what I've done and maybe you can explain to me where I'm going wrong, or show me how you would solve this equation, step by step; here is how I've done it ????
x^3 / x^2 -y^2 + y^3/ y^2 -x^2
= x^3/ (x-y)(x+y) + y^3/ (y-x)(y+x)
I figured (x+y) and (y+x) mean the same thing in the denominator; so I therefore did this:
x^3(y-x)/ (y-x)(x-y)(x+y) + y^3(x-y)/(y-x)(x-y)(x+y)
I then cancelled out the (y-x) from the numerator and denominator of the first expression and did the same with the second but cancelled out the (x-y) instead.
This then left me with:
x^3 +y^3/(x-y)(x+y)
I then factored out the top by sum of perfect cubes and got:
(x+y)(x^2-xy+y^2)/(x-y)(x+y)
I then cancelled out the x+y from the numerator and denominator and was left with this answer:
x^2-xy +y^2/ x-y

Hope it has helped by showing you how I got to this answer and any help you can give me would be greatly appreciated.

Answer 1

lets make it simple
I assume you omitted brackets copying from typeset form
Z = x^3/ ( x^2-y^2 )+ y^3/ (y^2-x^2 )
= x^3/ (x^2-y^2) - y^3/( x^2-y^2 )
= ( x^3 - y^3 ) / ( x^2 - y^2 )
= [ (x-y) ( x^2 + xy + y^2 ) ] / [ (x-y)(x+y) ]
= ( x^2 + xy + y^2 ) / (x + y )