Partial Fraction Decomposition with x and y in numerator and denominator

Question

I am trying to find the partial fraction decomposition of

$$\dfrac{x^2+y^2}{x^2-y^2}$$

I got it down to $x^2+y^2 = (A+B)x+(A-B)y$, but I cannot simplify it down anymore. Is this the wrong approach or am I not seeing something?

Answer 1

As a first step, you need to divide the polynomials so that the degree of the top (with respect to $x$) is less than the degree of the bottom. In this case, we have $$ \frac{x^2 + y^2}{x^2 - y^2} = \frac{(x^2-y^2) + 2y^2}{x^2 - y^2} = 1+\frac{2y^2}{x^2 - y^2} $$ Now, we set $$ \frac{2y^2}{x^2 - y^2} = \frac{A}{x-y} + \frac{B}{x+y} $$ To get $$ 2y^2 = (A+B)x + (A-B)y $$ Which gives us the system of equations $$ \begin{cases} A+B = 0\\ Ay-By = 2y^2 \end{cases} $$