Differential equation of 2nd order I have a differential equation $x''(t)=x(t)^2-x(t)$.
The exercise is as follows:
Let $x'(0)=0$. Then the solution $x(t)$ only depends on the initial position $x(0)$. 
Show that there is exactly one value of $x(0)$ for which the solution $x(t)$ is non-constant, yet tens to a finite value as $t$ tends to infinity. Calculate this $x(0)$, as well as the limiting value $x(+\infty)$. What happens to $x(t)$ as $t$ goes to infinity for other values of $x(0)$?
This is the question. My intention was to simply solve the reduced system
\begin{align*}
x'(t)&=y(t)\\
y'(t)&=x(t)^2-x(t)
\end{align*}
with barrows formula. But I'm not sure how to do that when the equations depend on each other. Any help on how to do this or maybe if there is an easier approach?
 A: Per the usual connection to physical systems, interpret this as motion generated by a force field, $\ddot x=-V'(x)$ with energy function $E(x,v)=\frac12v^2+V(x)$. From the given equation, 
$$
V(x)=\frac12 x^2 -\frac13 x^3
$$
can be identified. 
Because the values at the tangential points of the potential function are $V(0)=0$ and $V(1)=\frac16$, there is a minimum at zero, a valley of height $\frac16$  above it leading to oscillating solutions and above height $\frac16$ left-bounded, right-unbounded solutions.
Since the initial velocity is zero, the solutions all start on the graph of the potential function.

$E(x,\dot x)$ is a first integral of the differential equation, i.e., constant along trajectories. The level sets of $E$ in the $(x,v)$ plane give the curves that the trajectories follow. 
The potential $V(x)$ can be interpreted as a height field, a landscape with gravity. The solutions of the DE interpreted in this landscape give the motion of a frictionless, non-rolling ball or particle that follows this curve. Thus a motion starting inside the valley stays in the valley. Note that the valley includes points left from zero. Starting above the "hill" on the curve lets the ball climb above the hill after traversing the valley, then following the descent to negative infinity.
