Question

Prove that the system:

U_x = 2.999999x^2y + y
U_y = x^3 + x

has no solution at all.

Answer 1

Task:

Prove that the system:

$$u _ {x} = 2.999999x ^ {2} y + y$$ $$u _ {y} = x ^ {3} + x$$

has no solution at all.

Solution:

$$u _ {y} = x ^ {3} + x$$ $$u = x ^ {3} y + x y + f (x)$$ $$\left(x ^ {3} y + x y + f (x)\right) _ {x} = 2.999999x ^ {2} y + y$$ $$3 x ^ {2} y + y + f ^ {\prime} (x) = 2.999999x ^ {2} y + y$$ $$2.999999x ^ {2} y + y + \left(f ^ {\prime} (x) + 0.000001x ^ {2} y\right) = 2.999999x ^ {2} y + y$$ $$f ^ {\prime} (x) = 0$$

There is no analogy of $0.000001x^{2}y$ in the right side. There are no solutions