To simplify the given expression:
[tex]\[ \frac{x}{x-y} - \frac{x}{x+y} + \frac{2xy}{x^2 + y^2} \][/tex]
we'll proceed step-by-step.
First, we need to find a common denominator for the fractions. The common denominator for [tex]\(\frac{x}{x-y}\)[/tex] and [tex]\(\frac{x}{x+y}\)[/tex] is [tex]\((x-y)(x+y)\)[/tex], or equivalently, [tex]\(x^2 - y^2\)[/tex].
1. Rewrite the fractions with the common denominator [tex]\(x^2 - y^2\)[/tex]:
[tex]\[
\frac{x}{x-y} = \frac{x(x+y)}{(x-y)(x+y)} = \frac{x^2 + xy}{x^2 - y^2}
\][/tex]
[tex]\[
\frac{x}{x+y} = \frac{x(x-y)}{(x+y)(x-y)} = \frac{x^2 - xy}{x^2 - y^2}
\][/tex]
2. Now, combine the fractions:
[tex]\[
\frac{x}{x-y} - \frac{x}{x+y} = \frac{x^2 + xy}{x^2 - y^2} - \frac{x^2 - xy}{x^2 - y^2} = \frac{(x^2 + xy) - (x^2 - xy)}{x^2 - y^2}
\][/tex]
Simplify the numerator:
[tex]\[
(x^2 + xy) - (x^2 - xy) = x^2 + xy - x^2 + xy = 2xy
\][/tex]
Thus,
[tex]\[
\frac{x}{x-y} - \frac{x}{x+y} = \frac{2xy}{x^2 - y^2}
\][/tex]
3. Now, combine this result with the third term [tex]\(\frac{2xy}{x^2 + y^2}\)[/tex]:
[tex]\[
\frac{2xy}{x^2 - y^2} + \frac{2xy}{x^2 + y^2}
\][/tex]
4. We rewrite this as a single fraction by adding with a common denominator. The common denominator for these fractions is [tex]\((x^2 - y^2)(x^2 + y^2)\)[/tex]:
[tex]\[
\frac{2xy(x^2 + y^2) + 2xy(x^2 - y^2)}{(x^2 - y^2)(x^2 + y^2)}
\][/tex]
Since both terms in the numerator share a factor of [tex]\(2xy\)[/tex], we can factor that out:
[tex]\[
2xy \left( \frac{x^2 + y^2 + x^2 - y^2}{(x^2 - y^2)(x^2 + y^2)} \right)
\][/tex]
Simplify the numerator inside the parentheses:
[tex]\[
x^2 + y^2 + x^2 - y^2 = 2x^2
\][/tex]
So the fraction becomes:
[tex]\[
2xy \left( \frac{2x^2}{(x^2 - y^2)(x^2 + y^2)} \right)
\][/tex]
Combine the constants:
[tex]\[
\frac{4x^3y}{(x^2 - y^2)(x^2 + y^2)}
\][/tex]
Finally, observe that our denominator [tex]\((x^2 - y^2)(x^2 + y^2)\)[/tex]:
[tex]\[
(x^2 - y^2)(x^2 + y^2) = x^4 - y^4
\][/tex]
Thus the simplified expression is:
[tex]\[
\boxed{\frac{4x^3y}{x^4 - y^4}}
\][/tex]
