I am trying to solve the following system of nonlinear equations
$$ N(x,y) = (b-x) \cdot y - \int_0^x f(\zeta(\bar{x},y)) \, d\bar{x} = A $$
$$ M(x,y) = C \cdot (b - x) \cdot y - \left( D - \frac{\int_0^x f(\zeta(\bar{x},y)) \cdot \bar{x} \, d\bar{x} }{\int_0^x f(\zeta(\bar{x},y)) \, d\bar{x} } \right) \cdot \int_0^x f(\zeta(\bar{x},y)) \, d\bar{x} = B $$
$$f(\zeta(x,y)) = \frac{k \cdot \zeta - \zeta^2}{1+ (k-2) \zeta} $$
$$\zeta = -y \cdot x$$
It looks a tad intimidating, but the idea is the following:
- A, B, C, D, k and b are known constant values
- $\zeta$ is a function of $y$ and $x$
- The first and the last integrals represent an area under a certain curve
- The fraction of two integrals is the centroid of that area
My two unknowns are $x$ and $y$ and I would like to solve this system numerically using Python. However, since I cannot express this system of equations in terms of one variable (due to $f(\zeta)$ depending on $x$ and $y$), I do not know where I should start or which solution method to use.
Edit: this is not a question about implementation in Python or coding! I would like to know how I should go about numerically solving this system.
Edit2: so, I have managed to simplify the system. Now it looks like this
$$F \cdot (b-x) \cdot y+ G = \int_0^x \frac{k \cdot \zeta(\bar{x},y) - \zeta(\bar{x},y)^2}{1+ (k-2) \zeta(\bar{x},y)} d\bar{x}, \quad \zeta(x,y) = -x\cdot y$$
$F$ and $G$ are constants; $x$ and $y$ are unknowns.