# DSolve unevaluated. System of differential equations

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

• Didt you not get an error message: "DSolve::overdet: There are fewer dependent variables than equations, so the system is overdetermined." In fact, you can not solve 3 equations with only 2 variables: u and v Commented May 3, 2021 at 11:59
• Thank you for the comment. I do not get an error. The output is the same as the input. Could you check if my syntax is ok? I also tried to feed it only eq1 and eq2, getting the same (output = input). Commented May 3, 2021 at 12:04

Column[{eq1, eq2, eq3}]