The discussion centers on proving that the system of partial differential equations (PDEs) given by u_x - 2.999999x^2 y + y = 0 and u_y - x^3 + x = 0 has no solution. The approach taken involved integrating the first equation with respect to x and differentiating with respect to y, leading to the expression u_y - (2.999999/3)x^3 + x = f'(y). However, the conclusion drawn is that it is impossible to select a function f(y) that satisfies both equations simultaneously, confirming the non-existence of a solution.
PREREQUISITESMathematicians, students studying differential equations, and researchers in applied mathematics who are interested in the properties and behaviors of systems of PDEs.
perishingtardi said:Homework Statement
Show that there is no solution for the system
[tex]u_x - 2.999999x^2 y + y = 0,[/tex]
[tex]u_y - x^3 + x = 0.[/tex]
Homework Equations
The Attempt at a Solution
I took the first equation and integrated w.r.t x, then differentiated w.r.t y. But I'm not sure if it helps:
[tex]u_y - \frac{2.999999}{3}x^3 + x = f'(y)[/tex] where f(y) is an arbitrary function of y.