The second question was #9.2.17:

(a) Find an equation of the form $H(x,y) = c$ satisfied by the trajectories

$$ \frac{dx}{dt} = 2y, \qquad \frac{dy}{dt} = 8x. $$

(b) Plot several level curves of the function $H$. These are trajectories of the given system. Indicate the direction of motion on each trajectory.

First, we determine the function $H(x,y)$:

$$ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{8x}{2y} \Longleftrightarrow ydy = 4xdx \Longrightarrow H(x,y) = \frac{1}{2} y^2 - 2x^2 = c, $$

where $c$ is a constant of integration. For $c = -2,-1,0,1,2$, we have:

Therefore, for $c = 0$, the trajectories are two lines with slopes $2$ and $-2$ that intersect at the origin, and are separatrices. For $c \neq 0$, the trajectories are hyperbolÃ¦. In particular, for $c > 0$, the hyperbolÃ¦ lie along the ordinate; for $c < 0$, they lie along the abscissa. To determine the direction of the trajectories, we rewrite the system as the matrix equation

$$ \frac{d}{dt} \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{cc} 0 & 2 \\ 8 & 0 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right), $$

and we plug in the vectors $(x,y)^T=(0,1)$ and $(x,y)^T=(0,-1)$. This yields, respectively,

$$ \frac{d}{dt} \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} 2 \\ 0 \end{array} \right), \qquad \frac{d}{dt} \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} -2 \\ 0 \end{array} \right). $$

We conclude that the hyperbolÃ¦ along the ordinate are directed counter-clockwise. For the hyperbolÃ¦ along the abscissa, we can use $(x,y)^T=(1,0)$ and $(x,y)^T=(-1,0)$ to get, respectively,

$$ \frac{d}{dt} \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} 0 \\ 8 \end{array} \right), \qquad \frac{d}{dt} \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} 0 \\ -8 \end{array} \right). $$

Therefore, the hyperbolÃ¦ along the abscissa for $x < 0$ are directed downwards and those for $x > 0$ are directed upwards. This can be verified with a stream plot: