Is F(x,t) a Constant in the Given Dynamical System?

[Advent]

Homework Statement

Given the dynamical system $$\dot{x}=1-x^2$$, show that

$$F(x,t)=\frac{1+x}{1-x}e^{-2t}$$

is a constant of that system, and obtain the general solution of the differential equation with $$F(x,t)$$

Homework Equations

Above

The Attempt at a Solution

As $$F(x,t)$$ is a constant, then should satisfy

$$dF=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial t}dt=0$$

$$\frac{\partial F}{\partial x}=\frac{2(e^{-2t})}{(1-x)^2}$$

$$\frac{\partial F}{\partial t}=\frac{(1+x)(-2 e^{-2t})}{1-x}$$

Now, as $$\dot{x}=1-x^2$$

$$\frac{dx}{dt}=\frac{(1+x)e^{-2t}}{1-x} \frac{(1-x)^2}{e^{-2t}}=1-x^2$$

wich completes the proof.

Now, to compute the general solution $$x(t)$$ of the problem, should I use the fact that

$$\int\frac{\partial F}{\partial x}dx=-\int\frac{\partial F}{\partial t}dt$$

and use that

$$\frac{\partial F}{\partial x} \frac{\partial x}{\partial F}=1$$

to find an integral for $$x(t)$$.

Any kind of help is appreciated :D

 

[Jhero]

Thank you for your post. Your approach to proving that F(x,t) is a constant of the given dynamical system is correct. To obtain the general solution of the differential equation with F(x,t), you can use the fact that F(x,t) is a constant and substitute it into the equation \dot{x}=1-x^2. This will give you a new differential equation in terms of x and t. You can then use standard methods, such as separation of variables, to solve for x(t).

Alternatively, you can use the method of integrating factors to solve the differential equation with F(x,t). This involves multiplying both sides of the equation by an integrating factor, which is a function that depends only on t. In this case, the integrating factor would be e^{-2t}. This method may be more efficient in this case as it avoids the need for integration.

I hope this helps. Good luck with your problem!