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I have the task to sketch the solution of $$\begin{pmatrix} \dot x\\ \dot y \end{pmatrix} = \begin{pmatrix} y\\ -V'(x) \end{pmatrix},$$ where $V$ is a polynomial with $\lim\limits_{|x|\rightarrow \infty}V(x)= \infty$, two strict local minima $x_0<x_2$ and one strict local maxima $x_1 \in (x_0,x_2)$, where $V(x_2)< V(x_0)$. Does anyone know how to do this?

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This system has very particular form: you can rewrite it as $\ddot{x} = -V^\prime(x)$. Physicists instantly see this as a 1D particle subject to some conservative force with potential $V(x)$. The most important pecularity of such system is that it has energy conservation law: the quantity $\dfrac{\dot{x}^2(t)}{2} + V(x(t))$ which is full mechanical energy in this case is constant along any trajectory, i.e. $$\dfrac{\dot{x}^2(t)}{2} + V(x(t)) = \dfrac{\dot{x}^2(0)}{2} + V(x(0)).$$ Conservation of energy in such systems is a kind of inspiration for concept of "first integral". Just check that $\dfrac{y^2}{2} + V(x)$ doesn't change in time when you plug solution of ODE into it. This leads to one very useful observation: knowing how integral curves look like is the same as knowing how level sets of $\dfrac{y^2}{2} + V(x)$ look like. That's all you really need: just sketch the level sets of this function using remarks from the statement of your problem.

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