Consider a simple first-order differential equation like this, describing the evolution of the earth's atmospheric temperature according to a simple thermodynamic model including the albedo effect:
$$\frac{dx}{dt} = -\alpha x^4 + \beta(x) + \gamma$$
where $\beta(x)$ is a piecewise linear function.
There are three relevant equilibrium points, the two stable ones ($x_1, x_2$) being approached along different paths, depending solely on the initial value of $x$:
I wonder how for a given first-order differential equation like the one above a "energy" function $E(x)$ can be determined with local minima at $x_1$ and $x_2$ and a local maximum at $x_3$, as well as a "friction" function $f(x)$, such that "particles" starting at a given point $x_0$ with $\dot{x}_0 = 0$ driven by the "force"
$$F(t) = -k\frac{dE}{dx}(x(t)) - f(x(t))\dot{x}(t)$$
have the same solutions $x(t)$ as the differential equation.
The problem thus is: Given a simple first-order differential equation. Decide if there are functions $E(x)$, $f(x)$ that fulfill the requirement. If so, find them.
What's the name of this problem (or the general context it is posed in)? Which necessary and/or sufficient conditions on the differential equation are there for a solution to exist? Which exact solutions are there? Is there a systematic way to find an (approximate) solution?
To make things clear: For second-order differential equations, exact solutions are well known, e.g. $E(x) = x^2$, $f(x) = 0$ for the harmonic oscillator $\ddot{x} = -x$. This is just classical mechanics.