Question

Show that there exists a unique solution for the system:

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

together with the initial condition u(0,0) = 0.

Answer 1

U_x = 3x^2y+y
U_y = x^3+x
Integrate the first:
U = x^3 *y + y*x + f(y) where f(y) - some function.
Substitute it to the second equation:
(x^3 *y + y*x + f(y))_y = x^3+x
x^3 + x + f '(y) = x^3 + x
Therefore f '(y) = 0 => f(y) = const.
The initial condition U(0,0) = 0.
U = x^3 *y + y*x + const
U (0, 0) = const = 0
U (x, y) = x^3 *y + y*x

Answer: U (x, y) = x^3 *y + y*x.