# One dimensional mechanical system.

1. Nov 7, 2009

1. The problem statement, all variables and given/known data

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 $$F(x,t)$$

2. Relevant equations

Above

3. 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