# Smart variable change for non linear system

I'm trying to solve the following system of equations $$y=\sqrt{\epsilon-x^{2}}$$ $$y=\rho\frac{x}{\zeta-x}$$ for $x,y$, where $\epsilon>0,\rho>0,\zeta>0$ are real coefficients. The straightforward way is just by substitution, which leads to a forth order algebraic equation in one the variables (for example $x$), but the solution to this is really cumbersome (Mathematica) and hard to analyze.

This seems really symmetric so I was wondering if this can be solved as well by a smart change of variables (for example, transforming the system into a trascendental equation or using a complex plane transformation considering $x,y$ as complex variables) which may allow to express conveniently the solution in terms of some functions.

Thank you very much in advance

Best

## 1 Answer

I don't know if you have tried the usual transformation of $x$,$y$ to polar coordinates $r,\theta$. Your equations then take an (apparently) simpler form:

$$r = \epsilon$$ $$\tan\theta = \rho \frac{1}{\zeta-\epsilon\cos\theta}$$

Introducing the first inside the second you get a trascendental equation that may be easier to solve (at least numerically):

$$\zeta\tan\theta-\epsilon\sin\theta=\rho$$

Maybe you'll find a solution there :/

• Thank you very much! But now the question is how we solve $$A\tan\theta - B\sin\theta=1$$ analytically? I wonder an explicit solution in terms of special functions must exist no? – jdp89 Apr 14 '16 at 12:44