Solving a system of implicit functions

Question

I am trying to separate $u(x,y)$ and $v(x,y)$ in the given system:

\begin{equation} \begin{array}{cc} xu+yv=x+y \\ yu-xv=x-y \end{array} \end{equation}

However, when I arrange as matrix and solve for the determinants I get

$$\begin{vmatrix} xu & yv\\ yu & -xv \end{vmatrix} = \begin{vmatrix} x & y\\ x & -y \end{vmatrix}$$ $$\begin{equation} uv= \frac{2xy}{x^2+y^2} \end{equation}$$

But still, here I cannot separate $u$ and $v$. How can I do that?

Thanks!

Answer 1

Multiply the top equation by $y$ to get $xyu+y^2v=xy+y^2$, multiply the bottom by $x$ to get $xyu-x^2v=x^2-xy$. Subtract to get $$ v=\frac{-x^2+2xy+y^2}{x^2+y^2}\ . $$ Proceed similarly.

Answer 2

The matrix approach is to write the system as a linear system and use Cramer's rule: $$ \begin{pmatrix} x & y\\ y & -x \end{pmatrix} \begin{pmatrix} u\\ v \end{pmatrix} = \begin{pmatrix} x + y\\ x -y \end{pmatrix} $$

Here is another take, with complex numbers.

Let $z=u-iv$. Then $(x+iy)z=(x+y)+i(x-y)$ and it's easy to solve for $z$.