# Show system of PDEs has no solution

• perishingtardi
In summary, the conversation discusses a homework problem that requires showing that there is no solution for a system of equations involving u_x and u_y. The attempt at a solution involves integrating and differentiating the first equation, but it is unclear if this helps. The question also asks if it is possible to choose an arbitrary function f(y) to satisfy the second equation.

## Homework Statement

Show that there is no solution for the system
$$u_x - 2.999999x^2 y + y = 0,$$
$$u_y - x^3 + x = 0.$$

## 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:
$$u_y - \frac{2.999999}{3}x^3 + x = f'(y)$$ where f(y) is an arbitrary function of y.

perishingtardi said:

## Homework Statement

Show that there is no solution for the system
$$u_x - 2.999999x^2 y + y = 0,$$
$$u_y - x^3 + x = 0.$$

## 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:
$$u_y - \frac{2.999999}{3}x^3 + x = f'(y)$$ where f(y) is an arbitrary function of y.

And can you choose $f(y)$ such that $u_y - x^3 + x = 0$, as required?

## What is a PDE?

A PDE, or partial differential equation, is a type of mathematical equation that involves multiple variables and their partial derivatives. It is commonly used to model physical phenomena in fields such as physics, engineering, and finance.

## What does it mean for a system of PDEs to have no solution?

If a system of PDEs has no solution, it means that there is no combination of values for the variables that satisfy all of the equations in the system simultaneously. In other words, the equations are inconsistent and cannot be solved together.

## What are some possible reasons for a system of PDEs to have no solution?

There are several potential reasons for a system of PDEs to have no solution. It could be due to an error in the formulation of the equations, or the equations may be missing some necessary boundary conditions. It is also possible that the system is simply too complex to be solved analytically.

## Is it possible for a system of PDEs to have multiple solutions?

Yes, it is possible for a system of PDEs to have multiple solutions. In fact, in many real-world situations, there may be an infinite number of solutions that can satisfy the equations. This is known as the general solution, and it includes all possible solutions that could exist.

## How can it be determined if a system of PDEs has no solution?

To determine if a system of PDEs has no solution, one can use methods such as elimination, substitution, or graphing to solve the equations and see if they lead to any contradictions. Another approach is to use linear algebra techniques to determine if the system is consistent or inconsistent. In some cases, it may also be possible to prove that a system has no solution using mathematical techniques such as contradiction or proof by contradiction.

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