To simplify the given expression, we need to combine like terms and find a common denominator:

Ax + by/(a-b)(x+y) - bx - ay/(a+b)(x+y)
= Ax - bx + by/(a-b)(x+y) - ay/(a+b)(x+y)
= (A - b)x + by/(a-b)(x+y) - ay/(a+b)(x+y)

Finding a common denominator for the fractions, we get:

= [(A - b)(x)(a+b) + by(a+b) - ay(a-b)] / [(a-b)(a+b)(x+y)]
= [(Aa - Ab + bya + byb - aay + aay)] / [(a-b)(a+b)(x+y)]
= (Aa + byb - Ab) / [(a^2 - b^2)(x+y)]
= (a^2 + b^2) / (a^2 - b^2)

Thus, the simplified expression is (a^2 + b^2) / (a^2 - b^2).