DSolve unevaluated. System of differential equations

Question

I am trying to solve a system of differential equations with partial derivatives. The initial equation is just one

$$ \frac{(v-1)^3}{(a(u-1) u v +b (v-1))^2} \Big(\frac{\partial u}{\partial x} \frac{\partial v}{\partial y} - \frac{\partial v}{\partial x} \frac{\partial u}{\partial y} \Big) = 2 \frac{x^3 y^3 (x-1)}{(a(x-1) (y-1) + b x (1-xy))^2} $$

But since a and b are independent, this can be rearranged into 3 different equations: passing the denominators multiplying to each side, and finding the coefficient of $a^2$, $b^2$, and $a b$, we find the three different equations. However, when I feed it to DSolve, the input stays unevaluated... Perhaps Mathematica cannot solve this system?

eq1 = 2*(-1 + x)*x^3*y^3*(-1 + v[x, y])^2 == 
  x^2*(-1 + x*y)^2*(-1 + v[x, y])^3*(Derivative[0, 1][v][x, y]*
      Derivative[1, 0][u][x, y] - 
     Derivative[0, 1][u][x, y]*Derivative[1, 0][v][x, y])

eq2 = 2*(-1 + x)*x^3*y^3*(-1 + u[x, y])^2*u[x, y]^2*
   v[x, y]^2 == (-1 + x)^2*(-1 + y)^2*(-1 + 
      v[x, y])^3*(Derivative[0, 1][v][x, y]*
      Derivative[1, 0][u][x, y] - 
     Derivative[0, 1][u][x, y]*Derivative[1, 0][v][x, y])

eq3 = 4*(-1 + x)*x^3*y^3*(-1 + u[x, y])*u[x, y]*(-1 + v[x, y])*
   v[x, y] == -2*(-1 + x)*
   x*(-1 + y)*(-1 + 
     x*y)*(-1 + v[x, y])^3*(Derivative[0, 1][v][x, y]*
      Derivative[1, 0][u][x, y] - 
     Derivative[0, 1][u][x, y]*Derivative[1, 0][v][x, y])


DSolve[{eq1, eq2, eq3}, {u[x, y], v[x, y]}, {x, y}]

Any help or suggestion would be much appreciated

Answer 1

Rewrite your equations, so that all the derivatives are on the left side and then display them in a column:

Column[{eq1, eq2, eq3}]

You see that the left sides are all identical, but the right sides are different. That means, the equations contradict each other.

I think you must go back to where the equations come from and search for an error.