A function constant along an ODE's solution [closed]

Question

I need some help to understand a claim in a paper that I'm reading.

Let $v, z: I\subset \mathbb{R}\to \mathbb{R}$ solutions to the ODE $$\left\{ \begin{array}[cl]. v' &= h(z)\\ z' &= h(z) \tan v \end{array}\right.$$ Where the function $h: \mathbb{R} \to \mathbb{R}$ is a diffeomorphism and this is the only information about $h$.

I already know there exists these solutions. The paper says "$e^z\cos(v) $ is constant along the solutions of this ODE". What can mean "constant along a solution" in this case? And, how can this information help me to understand a solution of this ODE?

Answer 1

Let $f(t):=e^{z(t)}\cos v(t)$, where $v$ and $z$ are solutions to your ODE. Then $$f'(t)=e^{z(t)}z'(t)\cos v(t)-e^{z(t)}v'(t)\sin v(t) \\ =e^{z(t)}[h(z(t))\tan v(t)\,\cos v(t)-h(z(t))\sin v(t)]=0.$$ So, $f$ is constant, as desired.

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As for your question "And, how can this information help me to understand a solution of this ODE?": That $f(t)=e^{z(t)}\cos v(t)$ is a real constant $c$ means that, as $t$ varies, the point $(v(t),z(t))$ will be moving along the $c$-level curve $\{(V,Z)\in\mathbb R^2\colon e^Z\cos V=c\}$ of the function $(V,Z)\mapsto e^Z\cos V$. Parts of some of these level curves are shown here: