# Proving the Existence of a Constant for ODE Solutions and u(x,y)

• TheForumLord
In summary, the conversation discusses a given ODE and a function, and aims to prove that for each solution of the system, there exists a constant such that the function equals that constant for every t in the real numbers. The conversation also mentions the need to look at the derivative of the function and solving the ODE. In the end, the speaker states that they have managed to solve the problem.
TheForumLord

## Homework Statement

Given this ODE:

x' = x+y-xy^2
y' = -x-y+x^2y

and a function: u(x,y) = x^2+y^2-2ln|xy-1|

prove that for each soloution ( x(t), y(t) ) of this system, such as: x(t)*y(t) != 1 (doesn't equal...) , there exists a constsnt C such as: u ( x(t), y(t) ) = C for every t in R.

## The Attempt at a Solution

It's very clear that we need to look at the deriative of u... If it will be 0, then we'll get what we need...But since I haven't got that much knowledge in 2 variables functions, I can't really see what is the deriative of u, as well as how to solve this ODE...
So, I really need your help in:

1. Solving the ODE.
2. What is the deriative of u(t)?

TNX a lot!

You don't have to solve the ODE. You just have to find d/dt of u(x,y). Then substitute your expressions for dx/dt and dy/dt in and see if you get 0.

Yep...Inded...
I've managed to solve it...TNX a lot!

## 1. What is an ODE (Ordinary Differential Equation)?

An ODE is a type of mathematical equation that involves a function of one or more variables and its derivatives. It is used to model various physical phenomena and is commonly used in many fields of science and engineering.

## 2. Why is it important to prove the existence of a constant for ODE solutions?

Proving the existence of a constant for ODE solutions is important because it allows us to find a unique solution to the equation. This constant is known as the "arbitrary constant" and it represents the initial conditions of the system being modeled.

## 3. How is the existence of a constant for ODE solutions proven?

The existence of a constant for ODE solutions is proven using various mathematical techniques such as separation of variables, integrating factors, and substitution. These methods help to manipulate the equation and determine the value of the constant.

## 4. What role does the constant play in finding solutions to ODEs?

The constant plays a crucial role in finding solutions to ODEs as it represents the initial conditions of the system being modeled. It allows us to determine a specific solution that satisfies both the equation and the given initial conditions.

## 5. Is the constant in ODE solutions always unique?

No, the constant in ODE solutions is not always unique. It depends on the type of ODE being solved and the given initial conditions. In some cases, there may be multiple constants involved in the solution, and in others, there may be no constant at all.

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